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Theorem 3vfriswmgralem 28116
Description: Lemma for 3vfriswmgra 28117. (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 448 . . . . . . 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 383 . . . . . 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 3852 . . . . . . . . . 10  |-  ( w  =  A  ->  { A ,  w }  =  { A ,  A }
)
43eleq1d 2478 . . . . . . . . 9  |-  ( w  =  A  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
5 preq2 3852 . . . . . . . . . 10  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
65eleq1d 2478 . . . . . . . . 9  |-  ( w  =  B  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
74, 6rexprg 3826 . . . . . . . 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 978 . . . . . . 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 452 . . . . . 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 224 . . . . 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 2680 . . . . 5  |-  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <->  E. w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E ) )
1210, 11sylib 189 . . . 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 2927 . . . . . . . . 9  |-  w  e. 
_V
1413elpr 3800 . . . . . . . 8  |-  ( w  e.  { A ,  B }  <->  ( w  =  A  \/  w  =  B ) )
15 vex 2927 . . . . . . . . . . . 12  |-  y  e. 
_V
1615elpr 3800 . . . . . . . . . . 11  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
17 eqidd 2413 . . . . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . . . . 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 3852 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  { A ,  y }  =  { A ,  A }
)
2120eleq1d 2478 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
22 eqeq2 2421 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( A  =  y  <->  A  =  A ) )
2322imbi2d 308 . . . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . 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 21362 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
27 df-ne 2577 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  <->  -.  A  =  A )
28 eqid 2412 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  A  =  A
2928pm2.24i 138 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  A  =  A  ->  A  =  B )
3027, 29sylbi 188 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  =/=  A  ->  A  =  B )
3126, 30syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  B )
3231ex 424 . . . . . . . . . . . . . . . . . . . 20  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
33323ad2ant3 980 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3433adantr 452 . . . . . . . . . . . . . . . . . 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 29 . . . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . . . . 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 3852 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
3837eleq1d 2478 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
39 eqeq2 2421 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
4039imbi2d 308 . . . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . 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 369 . . . . . . . . . . . . . 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 2418 . . . . . . . . . . . . . . . . 17  |-  ( w  =  A  ->  (
w  =  y  <->  A  =  y ) )
4544imbi2d 308 . . . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 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 138 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  A  =  A  ->  B  =  A )
5027, 49sylbi 188 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  ->  B  =  A )
5126, 50syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  B  =  A )
5251ex 424 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
53523ad2ant3 980 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5453adantr 452 . . . . . . . . . . . . . . . . . . 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 29 . . . . . . . . . . . . . . . . . 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 23 . . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( B  =  y  <->  B  =  A ) )
5958imbi2d 308 . . . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . 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 2413 . . . . . . . . . . . . . . . . . 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 25 . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( B  =  y  <->  B  =  B ) )
6665imbi2d 308 . . . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . 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 369 . . . . . . . . . . . . . 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 2418 . . . . . . . . . . . . . . . . 17  |-  ( w  =  B  ->  (
w  =  y  <->  B  =  y ) )
7170imbi2d 308 . . . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 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 369 . . . . . . . . . . . 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 77 . . . . . . . . . . 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 188 . . . . . . . . . 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 419 . . . . . . . . 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 77 . . . . . . . 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 188 . . . . . . 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 422 . . . . . 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 29 . . . . 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 1639 . . . 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 2472 . . . . . 6  |-  ( w  =  y  ->  (
w  e.  { A ,  B }  <->  y  e.  { A ,  B }
) )
85 preq2 3852 . . . . . . 7  |-  ( w  =  y  ->  { A ,  w }  =  { A ,  y }
)
8685eleq1d 2478 . . . . . 6  |-  ( w  =  y  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  y }  e.  ran  E ) )
8784, 86anbi12d 692 . . . . 5  |-  ( w  =  y  ->  (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  <-> 
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) ) )
8887eu4 2301 . . . 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 646 . . 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 2681 . . 3  |-  ( E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
E! w ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
9189, 90sylibr 204 . 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 424 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 177    \/ wo 358    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2262    =/= wne 2575   E.wrex 2675   E!wreu 2676   {cpr 3783   {ctp 3784   class class class wbr 4180   ran crn 4846   USGrph cusg 21326
This theorem is referenced by:  3vfriswmgra  28117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-hash 11582  df-usgra 21328
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