Theorem fun2ssres 5931

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 5427	. . . 4 |- ( A C_ dom G -> ( ( F |` dom G ) |` A ) = ( F |` A ) )
2	1	eqcomd 2628	. . 3 |- ( A C_ dom G -> ( F |` A ) = ( ( F |` dom G ) |` A ) )
3		funssres 5930	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	reseq1d 5395	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) |` A ) = ( G |` A ) )
5	2, 4	sylan9eqr 2678	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
6	5	3impa 1259	1 |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   C_ wss 3574   dom cdm 5114   |` cres 5116   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890

This theorem is referenced by:  wfrlem12  7426  wfrlem14  7428  wfrlem17  7431  tfrlem9  7481  tfrlem9a  7482  tfrlem11  7484  bnj1503  30919  frrlem11  31792