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Theorem bnj1542 33395
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 5982 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
54necon3abid 2713 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  -. 
A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
6 df-ne 2664 . . . . . . 7  |-  ( ( F `  y )  =/=  ( G `  y )  <->  -.  ( F `  y )  =  ( G `  y ) )
76rexbii 2969 . . . . . 6  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  E. y  e.  A  -.  ( F `  y )  =  ( G `  y ) )
8 rexnal 2915 . . . . . 6  |-  ( E. y  e.  A  -.  ( F `  y )  =  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
97, 8bitri 249 . . . . 5  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
105, 9syl6bbr 263 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
112, 3, 10syl2anc 661 . . 3  |-  ( ph  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
121, 11mpbid 210 . 2  |-  ( ph  ->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) )
13 nfv 1683 . . 3  |-  F/ y ( F `  x
)  =/=  ( G `
 x )
14 bnj1542.4 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
1514nfcii 2619 . . . . 5  |-  F/_ x F
16 nfcv 2629 . . . . 5  |-  F/_ x
y
1715, 16nffv 5879 . . . 4  |-  F/_ x
( F `  y
)
18 nfcv 2629 . . . 4  |-  F/_ x
( G `  y
)
1917, 18nfne 2798 . . 3  |-  F/ x
( F `  y
)  =/=  ( G `
 y )
20 fveq2 5872 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
21 fveq2 5872 . . . 4  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
2220, 21neeq12d 2746 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( F `  y )  =/=  ( G `  y )
) )
2313, 19, 22cbvrex 3090 . 2  |-  ( E. x  e.  A  ( F `  x )  =/=  ( G `  x )  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y )
)
2412, 23sylibr 212 1  |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  bnj1523  33607
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