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Theorem bnj1253 29047
Description: Technical lemma for bnj60 29092. 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 28842 . . 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 29045 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
106bnj1292 28848 . . . . . . . . . . . 12  |-  D  C_  dom  g
11 fndm 5343 . . . . . . . . . . . 12  |-  ( g  Fn  d  ->  dom  g  =  d )
1210, 11syl5sseq 3226 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  D  C_  d )
13 fnssres 5357 . . . . . . . . . . 11  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
1412, 13mpdan 649 . . . . . . . . . 10  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
159, 14bnj31 28745 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
1615bnj1265 28845 . . . . . . . 8  |-  ( ph  ->  ( g  |`  D )  Fn  D )
173, 4, 5, 6, 7, 1, 8bnj1259 29046 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
186bnj1293 28849 . . . . . . . . . . . 12  |-  D  C_  dom  h
19 fndm 5343 . . . . . . . . . . . 12  |-  ( h  Fn  d  ->  dom  h  =  d )
2018, 19syl5sseq 3226 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  D  C_  d )
21 fnssres 5357 . . . . . . . . . . 11  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
2220, 21mpdan 649 . . . . . . . . . 10  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
2317, 22bnj31 28745 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
2423bnj1265 28845 . . . . . . . 8  |-  ( ph  ->  ( h  |`  D )  Fn  D )
25 ssid 3197 . . . . . . . . 9  |-  D  C_  D
26 fvreseq 5628 . . . . . . . . 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 652 . . . . . . . 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 642 . . . . . . 7  |-  ( ph  ->  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
29 residm 4986 . . . . . . . 8  |-  ( ( g  |`  D )  |`  D )  =  ( g  |`  D )
30 residm 4986 . . . . . . . 8  |-  ( ( h  |`  D )  |`  D )  =  ( h  |`  D )
3129, 30eqeq12i 2296 . . . . . . 7  |-  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
32 df-ral 2548 . . . . . . 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 278 . . . . . 6  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) ) ) )
34 fvres 5542 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( g `  x
) )
35 fvres 5542 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( h  |`  D ) `
 x )  =  ( h `  x
) )
3634, 35eqeq12d 2297 . . . . . . . 8  |-  ( x  e.  D  ->  (
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x )  <->  ( g `  x )  =  ( h `  x ) ) )
3736pm5.74i 236 . . . . . . 7  |-  ( ( x  e.  D  -> 
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) )  <->  ( x  e.  D  ->  ( g `
 x )  =  ( h `  x
) ) )
3837albii 1553 . . . . . 6  |-  ( A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) )
3933, 38syl6bb 252 . . . . 5  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) ) )
4039necon3abid 2479 . . . 4  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) ) )
41 df-rex 2549 . . . . 5  |-  ( E. x  e.  D  ( g `  x )  =/=  ( h `  x )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
42 pm4.61 415 . . . . . . 7  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
43 df-ne 2448 . . . . . . . 8  |-  ( ( g `  x )  =/=  ( h `  x )  <->  -.  (
g `  x )  =  ( h `  x ) )
4443anbi2i 675 . . . . . . 7  |-  ( ( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
4542, 44bitr4i 243 . . . . . 6  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
4645exbii 1569 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
47 exnal 1561 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) )
4841, 46, 473bitr2ri 265 . . . 4  |-  ( -. 
A. x ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
)
4940, 48syl6bb 252 . . 3  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
) )
502, 49mpbid 201 . 2  |-  ( ph  ->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
517neeq1i 2456 . . 3  |-  ( E  =/=  (/)  <->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
52 rabn0 3474 . . 3  |-  ( { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5351, 52bitri 240 . 2  |-  ( E  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5450, 53sylibr 203 1  |-  ( ph  ->  E  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28711    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1311  29054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-bnj17 28712
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