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Theorem bnj1542 28889
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1542.1  |-  ( ph  ->  F  Fn  A )
bnj1542.2  |-  ( ph  ->  G  Fn  A )
bnj1542.3  |-  ( ph  ->  F  =/=  G )
bnj1542.4  |-  ( w  e.  F  ->  A. x  w  e.  F )
Assertion
Ref Expression
bnj1542  |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
Distinct variable groups:    x, A    w, F    w, G, x
Allowed substitution hints:    ph( x, w)    A( w)    F( x)

Proof of Theorem bnj1542
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bnj1542.3 . . 3  |-  ( ph  ->  F  =/=  G )
2 bnj1542.1 . . . 4  |-  ( ph  ->  F  Fn  A )
3 bnj1542.2 . . . 4  |-  ( ph  ->  G  Fn  A )
4 eqfnfv 5622 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
54necon3abid 2479 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  -. 
A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
6 df-ne 2448 . . . . . . 7  |-  ( ( F `  y )  =/=  ( G `  y )  <->  -.  ( F `  y )  =  ( G `  y ) )
76rexbii 2568 . . . . . 6  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  E. y  e.  A  -.  ( F `  y )  =  ( G `  y ) )
8 rexnal 2554 . . . . . 6  |-  ( E. y  e.  A  -.  ( F `  y )  =  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
97, 8bitri 240 . . . . 5  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
105, 9syl6bbr 254 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
112, 3, 10syl2anc 642 . . 3  |-  ( ph  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
121, 11mpbid 201 . 2  |-  ( ph  ->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) )
13 nfv 1605 . . 3  |-  F/ y ( F `  x
)  =/=  ( G `
 x )
14 bnj1542.4 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
1514nfcii 2410 . . . . 5  |-  F/_ x F
16 nfcv 2419 . . . . 5  |-  F/_ x
y
1715, 16nffv 5532 . . . 4  |-  F/_ x
( F `  y
)
18 nfcv 2419 . . . 4  |-  F/_ x
( G `  y
)
1917, 18nfne 2539 . . 3  |-  F/ x
( F `  y
)  =/=  ( G `
 y )
20 fveq2 5525 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
21 fveq2 5525 . . . 4  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
2220, 21neeq12d 2461 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( F `  y )  =/=  ( G `  y )
) )
2313, 19, 22cbvrex 2761 . 2  |-  ( E. x  e.  A  ( F `  x )  =/=  ( G `  x )  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y )
)
2412, 23sylibr 203 1  |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  bnj1523  29101
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
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