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Theorem bnj1542 29228
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 5827 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
54necon3abid 2634 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  -. 
A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
6 df-ne 2601 . . . . . . 7  |-  ( ( F `  y )  =/=  ( G `  y )  <->  -.  ( F `  y )  =  ( G `  y ) )
76rexbii 2730 . . . . . 6  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  E. y  e.  A  -.  ( F `  y )  =  ( G `  y ) )
8 rexnal 2716 . . . . . 6  |-  ( E. y  e.  A  -.  ( F `  y )  =  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
97, 8bitri 241 . . . . 5  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  -.  A. y  e.  A  ( F `  y )  =  ( G `  y ) )
105, 9syl6bbr 255 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
112, 3, 10syl2anc 643 . . 3  |-  ( ph  ->  ( F  =/=  G  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) ) )
121, 11mpbid 202 . 2  |-  ( ph  ->  E. y  e.  A  ( F `  y )  =/=  ( G `  y ) )
13 nfv 1629 . . 3  |-  F/ y ( F `  x
)  =/=  ( G `
 x )
14 bnj1542.4 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
1514nfcii 2563 . . . . 5  |-  F/_ x F
16 nfcv 2572 . . . . 5  |-  F/_ x
y
1715, 16nffv 5735 . . . 4  |-  F/_ x
( F `  y
)
18 nfcv 2572 . . . 4  |-  F/_ x
( G `  y
)
1917, 18nfne 2695 . . 3  |-  F/ x
( F `  y
)  =/=  ( G `
 y )
20 fveq2 5728 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
21 fveq2 5728 . . . 4  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
2220, 21neeq12d 2616 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( F `  y )  =/=  ( G `  y )
) )
2313, 19, 22cbvrex 2929 . 2  |-  ( E. x  e.  A  ( F `  x )  =/=  ( G `  x )  <->  E. y  e.  A  ( F `  y )  =/=  ( G `  y )
)
2412, 23sylibr 204 1  |-  ( ph  ->  E. x  e.  A  ( F `  x )  =/=  ( G `  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  bnj1523  29440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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