Theorem funssfv 5587

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 5585	. . . 4 |- ( A e. dom G -> ( ( F |` dom G ) ` A ) = ( F ` A ) )
2	1	eqcomd 2202	. . 3 |- ( A e. dom G -> ( F ` A ) = ( ( F |` dom G ) ` A ) )
3		funssres 5301	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	fveq1d 5563	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) ` A ) = ( G ` A ) )
5	2, 4	sylan9eqr 2251	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )
6	5	3impa 1196	1 |- ( ( Fun F /\ G C_ F /\ A e. dom G ) -> ( F ` A ) = ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   C_ wss 3157   dom cdm 4664   |` cres 4666   Fun wfun 5253   ` cfv 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267

This theorem is referenced by:  tfrlem9  6386  tfrlemiubacc  6397  tfr1onlemubacc  6413  tfrcllemubacc  6426  ac6sfi  6968  ennnfonelemex  12656