Theorem fun2ssres 5298

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 4972	. . . 4 |- ( A C_ dom G -> ( ( F |` dom G ) |` A ) = ( F |` A ) )
2	1	eqcomd 2199	. . 3 |- ( A C_ dom G -> ( F |` A ) = ( ( F |` dom G ) |` A ) )
3		funssres 5297	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	reseq1d 4942	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) |` A ) = ( G |` A ) )
5	2, 4	sylan9eqr 2248	. 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 3154   dom cdm 4660   |` cres 4662   Fun wfun 5249

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-res 4672  df-fun 5257

This theorem is referenced by:  tfrlem9  6374  tfrlemiubacc  6385  tfr1onlemubacc  6401  tfrcllemubacc  6414