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Theorem bnj1253 29832
Description: Technical lemma for bnj60 29877. 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
bnj1253.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1253.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1253.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1253.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1253.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1253.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1253.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1253  |-  ( ph  ->  E  =/=  (/) )
Distinct variable groups:    A, f    B, f, g    B, h, f    D, d    x, D   
f, G, g    h, G    R, f    g, Y   
h, Y    f, d,
g    h, d    x, f, g    x, h
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, g, h, d)    B( x, y, d)    C( x, y, f, g, h, d)    D( y, f, g, h)    R( x, y, g, h, d)    E( x, y, f, g, h, d)    G( x, y, d)    Y( x, y, f, d)

Proof of Theorem bnj1253
StepHypRef Expression
1 bnj1253.6 . . . 4  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
21bnj1254 29627 . . 3  |-  ( ph  ->  ( g  |`  D )  =/=  ( h  |`  D ) )
3 bnj1253.1 . . . . . . . . . . 11  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4 bnj1253.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1253.3 . . . . . . . . . . 11  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 bnj1253.4 . . . . . . . . . . 11  |-  D  =  ( dom  g  i^i 
dom  h )
7 bnj1253.5 . . . . . . . . . . 11  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
8 bnj1253.7 . . . . . . . . . . 11  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
93, 4, 5, 6, 7, 1, 8bnj1256 29830 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
106bnj1292 29633 . . . . . . . . . . . 12  |-  D  C_  dom  g
11 fndm 5657 . . . . . . . . . . . 12  |-  ( g  Fn  d  ->  dom  g  =  d )
1210, 11syl5sseq 3448 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  D  C_  d )
13 fnssres 5671 . . . . . . . . . . 11  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
1412, 13mpdan 679 . . . . . . . . . 10  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
159, 14bnj31 29531 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
1615bnj1265 29630 . . . . . . . 8  |-  ( ph  ->  ( g  |`  D )  Fn  D )
173, 4, 5, 6, 7, 1, 8bnj1259 29831 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
186bnj1293 29634 . . . . . . . . . . . 12  |-  D  C_  dom  h
19 fndm 5657 . . . . . . . . . . . 12  |-  ( h  Fn  d  ->  dom  h  =  d )
2018, 19syl5sseq 3448 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  D  C_  d )
21 fnssres 5671 . . . . . . . . . . 11  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
2220, 21mpdan 679 . . . . . . . . . 10  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
2317, 22bnj31 29531 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
2423bnj1265 29630 . . . . . . . 8  |-  ( ph  ->  ( h  |`  D )  Fn  D )
25 ssid 3419 . . . . . . . . 9  |-  D  C_  D
26 fvreseq 5968 . . . . . . . . 9  |-  ( ( ( ( g  |`  D )  Fn  D  /\  ( h  |`  D )  Fn  D )  /\  D  C_  D )  -> 
( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
2725, 26mpan2 682 . . . . . . . 8  |-  ( ( ( g  |`  D )  Fn  D  /\  (
h  |`  D )  Fn  D )  ->  (
( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
2816, 24, 27syl2anc 671 . . . . . . 7  |-  ( ph  ->  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
29 residm 5114 . . . . . . . 8  |-  ( ( g  |`  D )  |`  D )  =  ( g  |`  D )
30 residm 5114 . . . . . . . 8  |-  ( ( h  |`  D )  |`  D )  =  ( h  |`  D )
3129, 30eqeq12i 2466 . . . . . . 7  |-  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
32 df-ral 2742 . . . . . . 7  |-  ( A. x  e.  D  (
( g  |`  D ) `
 x )  =  ( ( h  |`  D ) `  x
)  <->  A. x ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( ( h  |`  D ) `  x
) ) )
3328, 31, 323bitr3g 295 . . . . . 6  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) ) ) )
34 fvres 5862 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( g `  x
) )
35 fvres 5862 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( h  |`  D ) `
 x )  =  ( h `  x
) )
3634, 35eqeq12d 2467 . . . . . . . 8  |-  ( x  e.  D  ->  (
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x )  <->  ( g `  x )  =  ( h `  x ) ) )
3736pm5.74i 253 . . . . . . 7  |-  ( ( x  e.  D  -> 
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) )  <->  ( x  e.  D  ->  ( g `
 x )  =  ( h `  x
) ) )
3837albii 1695 . . . . . 6  |-  ( A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) )
3933, 38syl6bb 269 . . . . 5  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) ) )
4039necon3abid 2660 . . . 4  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) ) )
41 df-rex 2743 . . . . 5  |-  ( E. x  e.  D  ( g `  x )  =/=  ( h `  x )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
42 pm4.61 432 . . . . . . 7  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
43 df-ne 2624 . . . . . . . 8  |-  ( ( g `  x )  =/=  ( h `  x )  <->  -.  (
g `  x )  =  ( h `  x ) )
4443anbi2i 705 . . . . . . 7  |-  ( ( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
4542, 44bitr4i 260 . . . . . 6  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
4645exbii 1722 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
47 exnal 1703 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) )
4841, 46, 473bitr2ri 282 . . . 4  |-  ( -. 
A. x ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
)
4940, 48syl6bb 269 . . 3  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
) )
502, 49mpbid 215 . 2  |-  ( ph  ->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
517neeq1i 2688 . . 3  |-  ( E  =/=  (/)  <->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
52 rabn0 3720 . . 3  |-  ( { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5351, 52bitri 257 . 2  |-  ( E  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5450, 53sylibr 217 1  |-  ( ph  ->  E  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 986   A.wal 1446    = wceq 1448   E.wex 1667    e. wcel 1891   {cab 2438    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741    i^i cin 3371    C_ wss 3372   (/)c0 3699   <.cop 3942   class class class wbr 4374   dom cdm 4812    |` cres 4814    Fn wfn 5556   ` cfv 5561    /\ w-bnj17 29497    predc-bnj14 29499    FrSe w-bnj15 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-fv 5569  df-bnj17 29498
This theorem is referenced by:  bnj1311  29839
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