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Theorem bnj1502 28880
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1502.1  |-  ( ph  ->  Fun  F )
bnj1502.2  |-  ( ph  ->  G  C_  F )
bnj1502.3  |-  ( ph  ->  A  e.  dom  G
)
Assertion
Ref Expression
bnj1502  |-  ( ph  ->  ( F `  A
)  =  ( G `
 A ) )

Proof of Theorem bnj1502
StepHypRef Expression
1 bnj1502.1 . 2  |-  ( ph  ->  Fun  F )
2 bnj1502.2 . 2  |-  ( ph  ->  G  C_  F )
3 bnj1502.3 . 2  |-  ( ph  ->  A  e.  dom  G
)
4 funssfv 5543 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )
51, 2, 3, 4syl3anc 1182 1  |-  ( ph  ->  ( F `  A
)  =  ( G `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  bnj570  28937  bnj929  28968  bnj1450  29080  bnj1501  29097
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263
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