Theorem funssfv 5721

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 5719	. . . 4 |- ( A e. dom G -> ( ( F |` dom G ) ` A ) = ( F ` A ) )
2	1	eqcomd 2244	. . 3 |- ( A e. dom G -> ( F ` A ) = ( ( F |` dom G ) ` A ) )
3		funssres 5420	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	fveq1d 5697	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) ` A ) = ( G ` A ) )
5	2, 4	sylan9eqr 2293	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )
6	5	3impa 1225	1 |- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   dom cdm 4774   |` cres 4776   Fun wfun 5371   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385

This theorem is used by:  funsssuppss  6498  tfrlem9  6590  tfrlemiubacc  6601  tfr1onlemubacc  6617  tfrcllemubacc  6630  ac6sfi  7202  ennnfonelemex  13305  subgruhgredgdm  16511  subumgredg2en  16512  subupgr  16514