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Theorem bnj1253 32321
Description: Technical lemma for bnj60 32366. 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 32116 . . 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 32319 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
106bnj1292 32122 . . . . . . . . . . . 12  |-  D  C_  dom  g
11 fndm 5613 . . . . . . . . . . . 12  |-  ( g  Fn  d  ->  dom  g  =  d )
1210, 11syl5sseq 3507 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  D  C_  d )
13 fnssres 5627 . . . . . . . . . . 11  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
1412, 13mpdan 668 . . . . . . . . . 10  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
159, 14bnj31 32021 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
1615bnj1265 32119 . . . . . . . 8  |-  ( ph  ->  ( g  |`  D )  Fn  D )
173, 4, 5, 6, 7, 1, 8bnj1259 32320 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
186bnj1293 32123 . . . . . . . . . . . 12  |-  D  C_  dom  h
19 fndm 5613 . . . . . . . . . . . 12  |-  ( h  Fn  d  ->  dom  h  =  d )
2018, 19syl5sseq 3507 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  D  C_  d )
21 fnssres 5627 . . . . . . . . . . 11  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
2220, 21mpdan 668 . . . . . . . . . 10  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
2317, 22bnj31 32021 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
2423bnj1265 32119 . . . . . . . 8  |-  ( ph  ->  ( h  |`  D )  Fn  D )
25 ssid 3478 . . . . . . . . 9  |-  D  C_  D
26 fvreseq 5909 . . . . . . . . 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 671 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
29 residm 5244 . . . . . . . 8  |-  ( ( g  |`  D )  |`  D )  =  ( g  |`  D )
30 residm 5244 . . . . . . . 8  |-  ( ( h  |`  D )  |`  D )  =  ( h  |`  D )
3129, 30eqeq12i 2472 . . . . . . 7  |-  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
32 df-ral 2801 . . . . . . 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 287 . . . . . 6  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) ) ) )
34 fvres 5808 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( g `  x
) )
35 fvres 5808 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( h  |`  D ) `
 x )  =  ( h `  x
) )
3634, 35eqeq12d 2474 . . . . . . . 8  |-  ( x  e.  D  ->  (
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x )  <->  ( g `  x )  =  ( h `  x ) ) )
3736pm5.74i 245 . . . . . . 7  |-  ( ( x  e.  D  -> 
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) )  <->  ( x  e.  D  ->  ( g `
 x )  =  ( h `  x
) ) )
3837albii 1611 . . . . . 6  |-  ( A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) )
3933, 38syl6bb 261 . . . . 5  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) ) )
4039necon3abid 2695 . . . 4  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) ) )
41 df-rex 2802 . . . . 5  |-  ( E. x  e.  D  ( g `  x )  =/=  ( h `  x )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
42 pm4.61 426 . . . . . . 7  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
43 df-ne 2647 . . . . . . . 8  |-  ( ( g `  x )  =/=  ( h `  x )  <->  -.  (
g `  x )  =  ( h `  x ) )
4443anbi2i 694 . . . . . . 7  |-  ( ( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
4542, 44bitr4i 252 . . . . . 6  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
4645exbii 1635 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
47 exnal 1619 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) )
4841, 46, 473bitr2ri 274 . . . 4  |-  ( -. 
A. x ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
)
4940, 48syl6bb 261 . . 3  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
) )
502, 49mpbid 210 . 2  |-  ( ph  ->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
517neeq1i 2734 . . 3  |-  ( E  =/=  (/)  <->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
52 rabn0 3760 . . 3  |-  ( { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5351, 52bitri 249 . 2  |-  ( E  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5450, 53sylibr 212 1  |-  ( ph  ->  E  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2437    =/= wne 2645   A.wral 2796   E.wrex 2797   {crab 2800    i^i cin 3430    C_ wss 3431   (/)c0 3740   <.cop 3986   class class class wbr 4395   dom cdm 4943    |` cres 4945    Fn wfn 5516   ` cfv 5521    /\ w-bnj17 31987    predc-bnj14 31989    FrSe w-bnj15 31993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529  df-bnj17 31988
This theorem is referenced by:  bnj1311  32328
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