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Theorem bnj1536 28886
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1536.1  |-  ( ph  ->  F  Fn  A )
bnj1536.2  |-  ( ph  ->  G  Fn  A )
bnj1536.3  |-  ( ph  ->  B  C_  A )
bnj1536.4  |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
Assertion
Ref Expression
bnj1536  |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem bnj1536
StepHypRef Expression
1 bnj1536.4 . 2  |-  ( ph  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
2 bnj1536.1 . . 3  |-  ( ph  ->  F  Fn  A )
3 bnj1536.2 . . 3  |-  ( ph  ->  G  Fn  A )
4 bnj1536.3 . . 3  |-  ( ph  ->  B  C_  A )
5 fvreseq 5628 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
62, 3, 4, 5syl21anc 1181 . 2  |-  ( ph  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
71, 6mpbird 223 1  |-  ( ph  ->  ( F  |`  B )  =  ( G  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   A.wral 2543    C_ wss 3152    |` cres 4691    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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