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Theorem bnj1502 29196
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 5559 . 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 1632    e. wcel 1696    C_ wss 3165   dom cdm 4705   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  bnj570  29253  bnj929  29284  bnj1450  29396  bnj1501  29413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279
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