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Theorem bnj1311 29370
Description: Technical lemma for bnj60 29408. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1311.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1311.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1311.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1311.4  |-  D  =  ( dom  g  i^i 
dom  h )
Assertion
Ref Expression
bnj1311  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Distinct variable groups:    A, d,
f, x    B, f,
g    B, h, f    D, d, x    G, d, f, g    h, G, d    R, d, f, x    g, Y    h, Y    x, g    x, h
Allowed substitution hints:    A( g, h)    B( x, d)    C( x, f, g, h, d)    D( f, g, h)    R( g, h)    G( x)    Y( x, f, d)

Proof of Theorem bnj1311
Dummy variables  w  z  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 227 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  (
h  |`  D ) ) )
21bnj1232 29152 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  R  FrSe  A )
3 ssrab2 3271 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  C_  D
4 bnj1311.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
51bnj1235 29153 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
g  e.  C )
6 bnj1311.2 . . . . . . . . . . . 12  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
7 bnj1311.3 . . . . . . . . . . . 12  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
8 eqid 2296 . . . . . . . . . . . 12  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
9 eqid 2296 . . . . . . . . . . . 12  |-  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) }  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
106, 7, 8, 9bnj1234 29359 . . . . . . . . . . 11  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
115, 10syl6eleq 2386 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) } )
12 abid 2284 . . . . . . . . . . . . . 14  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  <->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) )
1312bnj1238 29155 . . . . . . . . . . . . 13  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d  e.  B  g  Fn  d )
1413bnj1196 29143 . . . . . . . . . . . 12  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d
( d  e.  B  /\  g  Fn  d
) )
15 bnj1311.1 . . . . . . . . . . . . . . 15  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
1615abeq2i 2403 . . . . . . . . . . . . . 14  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1716simplbi 446 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
18 fndm 5359 . . . . . . . . . . . . 13  |-  ( g  Fn  d  ->  dom  g  =  d )
1917, 18bnj1241 29156 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  dom  g  C_  A
)
2014, 19bnj593 29090 . . . . . . . . . . 11  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d dom  g  C_  A )
2120bnj937 29119 . . . . . . . . . 10  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  dom  g  C_  A )
22 ssinss1 3410 . . . . . . . . . 10  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
2311, 21, 223syl 18 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
( dom  g  i^i  dom  h )  C_  A
)
244, 23syl5eqss 3235 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  D  C_  A )
253, 24syl5ss 3203 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  C_  A )
26 eqid 2296 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  =  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }
27 biid 227 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  <->  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  /\  A. y  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x ) )
2815, 6, 7, 4, 26, 1, 27bnj1253 29363 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
29 nfrab1 2733 . . . . . . . . 9  |-  F/_ x { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029nfcrii 2425 . . . . . . . 8  |-  ( z  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  A. x  z  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
3130bnj1228 29357 . . . . . . 7  |-  ( ( R  FrSe  A  /\  { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) } 
C_  A  /\  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/) )  ->  E. x  e.  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) } A. y  e.  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  -.  y R x )
322, 25, 28, 31syl3anc 1182 . . . . . 6  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  E. x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } A. y  e.  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x )
33 ax-17 1606 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  R  FrSe  A )
3415bnj1309 29368 . . . . . . . . . 10  |-  ( w  e.  B  ->  A. x  w  e.  B )
357, 34bnj1307 29369 . . . . . . . . 9  |-  ( w  e.  C  ->  A. x  w  e.  C )
3635nfcii 2423 . . . . . . . 8  |-  F/_ x C
3736nfcrii 2425 . . . . . . 7  |-  ( g  e.  C  ->  A. x  g  e.  C )
3836nfcrii 2425 . . . . . . 7  |-  ( h  e.  C  ->  A. x  h  e.  C )
39 ax-17 1606 . . . . . . 7  |-  ( ( g  |`  D )  =/=  ( h  |`  D )  ->  A. x ( g  |`  D )  =/=  (
h  |`  D ) )
4033, 37, 38, 39bnj982 29126 . . . . . 6  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  A. x ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
4132, 27, 40bnj1521 29199 . . . . 5  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  E. x ( ( R 
FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  /\  A. y  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x ) )
42 simp2 956 . . . . 5  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  ->  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
4315, 6, 7, 4, 26, 1, 27bnj1279 29364 . . . . . . . . 9  |-  ( ( x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) } )  =  (/) )
44433adant1 973 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) } )  =  (/) )
4515, 6, 7, 4, 26, 1, 27, 44bnj1280 29366 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
46 eqid 2296 . . . . . . 7  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
47 eqid 2296 . . . . . . 7  |-  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x
)  =  ( G `
 <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >. )
) }  =  {
h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
) ) }
4815, 6, 7, 4, 26, 1, 27, 45, 8, 9, 46, 47bnj1296 29367 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
( g `  x
)  =  ( h `
 x ) )
4926bnj1538 29203 . . . . . . 7  |-  ( x  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  (
g `  x )  =/=  ( h `  x
) )
5049necon2bi 2505 . . . . . 6  |-  ( ( g `  x )  =  ( h `  x )  ->  -.  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
5148, 50syl 15 . . . . 5  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  ->  -.  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
5241, 42, 51bnj1304 29168 . . . 4  |-  -.  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )
53 df-bnj17 29028 . . . 4  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  <->  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  /\  (
g  |`  D )  =/=  ( h  |`  D ) ) )
5452, 53mtbi 289 . . 3  |-  -.  (
( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
)  /\  ( g  |`  D )  =/=  (
h  |`  D ) )
5554imnani 412 . 2  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  -.  ( g  |`  D )  =/=  (
h  |`  D ) )
56 nne 2463 . 2  |-  ( -.  ( g  |`  D )  =/=  ( h  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
5755, 56sylib 188 1  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj1326  29372  bnj60  29408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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