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Theorem 3vfriswmgralem 28456
Description: Lemma for 3vfriswmgra 28457. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
3vfriswmgralem  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Distinct variable groups:    w, A    w, B    w, C    w, E    w, X    w, Y

Proof of Theorem 3vfriswmgralem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr 449 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  { A ,  B }  e.  ran  E )
21olcd 384 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) )
3 preq2 3886 . . . . . . . . . 10  |-  ( w  =  A  ->  { A ,  w }  =  { A ,  A }
)
43eleq1d 2504 . . . . . . . . 9  |-  ( w  =  A  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
5 preq2 3886 . . . . . . . . . 10  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
65eleq1d 2504 . . . . . . . . 9  |-  ( w  =  B  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
74, 6rexprg 3860 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  { A ,  w }  e.  ran  E  <->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
873ad2ant1 979 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
98adantr 453 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
102, 9mpbird 225 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
11 df-rex 2713 . . . . 5  |-  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <->  E. w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E ) )
1210, 11sylib 190 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
13 vex 2961 . . . . . . . . 9  |-  w  e. 
_V
1413elpr 3834 . . . . . . . 8  |-  ( w  e.  { A ,  B }  <->  ( w  =  A  \/  w  =  B ) )
15 vex 2961 . . . . . . . . . . . 12  |-  y  e. 
_V
1615elpr 3834 . . . . . . . . . . 11  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
17 eqidd 2439 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A )
1817a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) )
1918a1ii 26 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
20 preq2 3886 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  { A ,  y }  =  { A ,  A }
)
2120eleq1d 2504 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
22 eqeq2 2447 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( A  =  y  <->  A  =  A ) )
2322imbi2d 309 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) )
2423imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
2519, 21, 243imtr4d 261 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
26 usgraedgrn 21403 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
27 df-ne 2603 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  <->  -.  A  =  A )
28 eqid 2438 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  A  =  A
2928pm2.24i 139 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  A  =  A  ->  A  =  B )
3027, 29sylbi 189 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  =/=  A  ->  A  =  B )
3126, 30syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  B )
3231ex 425 . . . . . . . . . . . . . . . . . . . 20  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
33323ad2ant3 981 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3433adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3534com12 30 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) )
3635a1ii 26 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
37 preq2 3886 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
3837eleq1d 2504 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
39 eqeq2 2447 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
4039imbi2d 309 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) )
4140imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
4236, 38, 413imtr4d 261 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4325, 42jaoi 370 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
44 eqeq1 2444 . . . . . . . . . . . . . . . . 17  |-  ( w  =  A  ->  (
w  =  y  <->  A  =  y ) )
4544imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( w  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) )
464, 45imbi12d 313 . . . . . . . . . . . . . . 15  |-  ( w  =  A  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4746imbi2d 309 . . . . . . . . . . . . . 14  |-  ( w  =  A  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) ) )
4843, 47syl5ibr 214 . . . . . . . . . . . . 13  |-  ( w  =  A  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
4928pm2.24i 139 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  A  =  A  ->  B  =  A )
5027, 49sylbi 189 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  ->  B  =  A )
5126, 50syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  B  =  A )
5251ex 425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
53523ad2ant3 981 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5453adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5554com12 30 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) )
5655a1d 24 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
5756a1i 11 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
58 eqeq2 2447 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( B  =  y  <->  B  =  A ) )
5958imbi2d 309 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
6059imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
6157, 21, 603imtr4d 261 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
62 eqidd 2439 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B )
6362a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) )
6463a1ii 26 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
65 eqeq2 2447 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( B  =  y  <->  B  =  B ) )
6665imbi2d 309 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) )
6766imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
6864, 38, 673imtr4d 261 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
6961, 68jaoi 370 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
70 eqeq1 2444 . . . . . . . . . . . . . . . . 17  |-  ( w  =  B  ->  (
w  =  y  <->  B  =  y ) )
7170imbi2d 309 . . . . . . . . . . . . . . . 16  |-  ( w  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) )
726, 71imbi12d 313 . . . . . . . . . . . . . . 15  |-  ( w  =  B  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
7372imbi2d 309 . . . . . . . . . . . . . 14  |-  ( w  =  B  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) ) )
7469, 73syl5ibr 214 . . . . . . . . . . . . 13  |-  ( w  =  B  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7548, 74jaoi 370 . . . . . . . . . . . 12  |-  ( ( w  =  A  \/  w  =  B )  ->  ( ( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7675com3l 78 . . . . . . . . . . 11  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7716, 76sylbi 189 . . . . . . . . . 10  |-  ( y  e.  { A ,  B }  ->  ( { A ,  y }  e.  ran  E  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7877imp 420 . . . . . . . . 9  |-  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
7978com3l 78 . . . . . . . 8  |-  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8014, 79sylbi 189 . . . . . . 7  |-  ( w  e.  { A ,  B }  ->  ( { A ,  w }  e.  ran  E  ->  (
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
)  ->  ( (
( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8180imp31 423 . . . . . 6  |-  ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )
8281com12 30 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  w  =  y ) )
8382alrimivv 1643 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) )
84 eleq1 2498 . . . . . 6  |-  ( w  =  y  ->  (
w  e.  { A ,  B }  <->  y  e.  { A ,  B }
) )
85 preq2 3886 . . . . . . 7  |-  ( w  =  y  ->  { A ,  w }  =  { A ,  y }
)
8685eleq1d 2504 . . . . . 6  |-  ( w  =  y  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  y }  e.  ran  E ) )
8784, 86anbi12d 693 . . . . 5  |-  ( w  =  y  ->  (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  <-> 
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) ) )
8887eu4 2322 . . . 4  |-  ( E! w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E )  <->  ( E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) ) )
8912, 83, 88sylanbrc 647 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
90 df-reu 2714 . . 3  |-  ( E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
E! w ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
9189, 90sylibr 205 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
9291ex 425 1  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283    =/= wne 2601   E.wrex 2708   E!wreu 2709   {cpr 3817   {ctp 3818   class class class wbr 4214   ran crn 4881   USGrph cusg 21367
This theorem is referenced by:  3vfriswmgra  28457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-hash 11621  df-usgra 21369
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