Theorem fun2ssres 5301

Description: Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
fun2ssres	|- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )

Proof of Theorem fun2ssres

Step	Hyp	Ref	Expression
1		resabs1 4975	. . . 4 |- ( A C_ dom G -> ( ( F |` dom G ) |` A ) = ( F |` A ) )
2	1	eqcomd 2202	. . 3 |- ( A C_ dom G -> ( F |` A ) = ( ( F |` dom G ) |` A ) )
3		funssres 5300	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	reseq1d 4945	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) |` A ) = ( G |` A ) )
5	2, 4	sylan9eqr 2251	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
6	5	3impa 1196	1 |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   C_ wss 3157   dom cdm 4663   |` cres 4665   Fun wfun 5252

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 4151  ax-pow 4207  ax-pr 4242

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-fun 5260

This theorem is referenced by:  tfrlem9  6377  tfrlemiubacc  6388  tfr1onlemubacc  6404  tfrcllemubacc  6417