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Theorem bnj1542 32205
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 5909 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
54necon3abid 2698 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  -. 
A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
6 df-ne 2650 . . . . . . 7  |-  ( ( F `  y )  =/=  ( G `  y )  <->  -.  ( F `  y )  =  ( G `  y ) )
76rexbii 2862 . . . . . 6  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  E. y  e.  A  -.  ( F `  y )  =  ( G `  y ) )
8 rexnal 2854 . . . . . 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 1674 . . 3  |-  F/ y ( F `  x
)  =/=  ( G `
 x )
14 bnj1542.4 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
1514nfcii 2606 . . . . 5  |-  F/_ x F
16 nfcv 2616 . . . . 5  |-  F/_ x
y
1715, 16nffv 5809 . . . 4  |-  F/_ x
( F `  y
)
18 nfcv 2616 . . . 4  |-  F/_ x
( G `  y
)
1917, 18nfne 2783 . . 3  |-  F/ x
( F `  y
)  =/=  ( G `
 y )
20 fveq2 5802 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
21 fveq2 5802 . . . 4  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
2220, 21neeq12d 2731 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( F `  y )  =/=  ( G `  y )
) )
2313, 19, 22cbvrex 3050 . 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 1368    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    Fn wfn 5524   ` cfv 5529
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537
This theorem is referenced by:  bnj1523  32417
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