Theorem funssfv 5696

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 5694	. . . 4 |- ( A e. dom G -> ( ( F |` dom G ) ` A ) = ( F ` A ) )
2	1	eqcomd 2238	. . 3 |- ( A e. dom G -> ( F ` A ) = ( ( F |` dom G ) ` A ) )
3		funssres 5395	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	fveq1d 5672	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) ` A ) = ( G ` A ) )
5	2, 4	sylan9eqr 2287	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )
6	5	3impa 1221	1 |- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   dom cdm 4749   |` cres 4751   Fun wfun 5346   ` cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360

This theorem is referenced by:  funsssuppss  6458  tfrlem9  6550  tfrlemiubacc  6561  tfr1onlemubacc  6577  tfrcllemubacc  6590  ac6sfi  7155  ennnfonelemex  13165  subgruhgredgdm  16265  subumgredg2en  16266  subupgr  16268