Theorem fun2ssres 5241

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 4920	. . . 4 |- ( A C_ dom G -> ( ( F |` dom G ) |` A ) = ( F |` A ) )
2	1	eqcomd 2176	. . 3 |- ( A C_ dom G -> ( F |` A ) = ( ( F |` dom G ) |` A ) )
3		funssres 5240	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	reseq1d 4890	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) |` A ) = ( G |` A ) )
5	2, 4	sylan9eqr 2225	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
6	5	3impa 1189	1 |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   C_ wss 3121   dom cdm 4611   |` cres 4613   Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-fun 5200

This theorem is referenced by:  tfrlem9  6298  tfrlemiubacc  6309  tfr1onlemubacc  6325  tfrcllemubacc  6338