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Theorem bnj1503 28560
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1503.1  |-  ( ph  ->  Fun  F )
bnj1503.2  |-  ( ph  ->  G  C_  F )
bnj1503.3  |-  ( ph  ->  A  C_  dom  G )
Assertion
Ref Expression
bnj1503  |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )

Proof of Theorem bnj1503
StepHypRef Expression
1 bnj1503.1 . 2  |-  ( ph  ->  Fun  F )
2 bnj1503.2 . 2  |-  ( ph  ->  G  C_  F )
3 bnj1503.3 . 2  |-  ( ph  ->  A  C_  dom  G )
4 fun2ssres 5436 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A ) )
51, 2, 3, 4syl3anc 1184 1  |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3265   dom cdm 4820    |` cres 4822   Fun wfun 5390
This theorem is referenced by:  bnj1442  28758  bnj1450  28759  bnj1501  28776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-fun 5398
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