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Theorem bnj1542 34335
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 5957 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
54necon3abid 2700 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =/=  G  <->  -. 
A. y  e.  A  ( F `  y )  =  ( G `  y ) ) )
6 df-ne 2651 . . . . . . 7  |-  ( ( F `  y )  =/=  ( G `  y )  <->  -.  ( F `  y )  =  ( G `  y ) )
76rexbii 2956 . . . . . 6  |-  ( E. y  e.  A  ( F `  y )  =/=  ( G `  y )  <->  E. y  e.  A  -.  ( F `  y )  =  ( G `  y ) )
8 rexnal 2902 . . . . . 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 659 . . 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 1712 . . 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 5855 . . . 4  |-  F/_ x
( F `  y
)
18 nfcv 2616 . . . 4  |-  F/_ x
( G `  y
)
1917, 18nfne 2785 . . 3  |-  F/ x
( F `  y
)  =/=  ( G `
 y )
20 fveq2 5848 . . . 4  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
21 fveq2 5848 . . . 4  |-  ( x  =  y  ->  ( G `  x )  =  ( G `  y ) )
2220, 21neeq12d 2733 . . 3  |-  ( x  =  y  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( F `  y )  =/=  ( G `  y )
) )
2313, 19, 22cbvrex 3078 . 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 367   A.wal 1396    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  bnj1523  34547
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