Theorem funssfv 5520

Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Assertion

Ref	Expression
funssfv	|- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )

Proof of Theorem funssfv

Step	Hyp	Ref	Expression
1		fvres 5518	. . . 4 |- ( A e. dom G -> ( ( F |` dom G ) ` A ) = ( F ` A ) )
2	1	eqcomd 2176	. . 3 |- ( A e. dom G -> ( F ` A ) = ( ( F |` dom G ) ` A ) )
3		funssres 5238	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	fveq1d 5496	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) ` A ) = ( G ` A ) )
5	2, 4	sylan9eqr 2225	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )
6	5	3impa 1189	1 |- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   C_ wss 3121   dom cdm 4609   |` cres 4611   Fun wfun 5190   ` cfv 5196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-res 4621  df-iota 5158  df-fun 5198  df-fv 5204

This theorem is referenced by:  tfrlem9  6295  tfrlemiubacc  6306  tfr1onlemubacc  6322  tfrcllemubacc  6335  ac6sfi  6872  ennnfonelemex  12356