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Theorem bnj1503 29121
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 5486 . 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 1652    C_ wss 3312   dom cdm 4870    |` cres 4872   Fun wfun 5440
This theorem is referenced by:  bnj1442  29319  bnj1450  29320  bnj1501  29337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-fun 5448
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