Theorem bnj1503 30919

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

Step	Hyp	Ref	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 5931	. 2 |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
5	1, 2, 3, 4	syl3anc 1326	1 |- ( ph -> ( F |` A ) = ( G |` A ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   dom cdm 5114   |` cres 5116   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890

This theorem is referenced by:  bnj1442  31117  bnj1450  31118  bnj1501  31135