Theorem fun2ssres 5231

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 4913	. . . 4 |- ( A C_ dom G -> ( ( F |` dom G ) |` A ) = ( F |` A ) )
2	1	eqcomd 2171	. . 3 |- ( A C_ dom G -> ( F |` A ) = ( ( F |` dom G ) |` A ) )
3		funssres 5230	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
4	3	reseq1d 4883	. . 3 |- ( ( Fun F /\ G C_ F ) -> ( ( F |` dom G ) |` A ) = ( G |` A ) )
5	2, 4	sylan9eqr 2221	. 2 |- ( ( ( Fun F /\ G C_ F ) /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )
6	5	3impa 1184	1 |- ( ( Fun F /\ G C_ F /\ A C_ dom G ) -> ( F |` A ) = ( G |` A ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   C_ wss 3116   dom cdm 4604   |` cres 4606   Fun wfun 5182

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-fun 5190

This theorem is referenced by:  tfrlem9  6287  tfrlemiubacc  6298  tfr1onlemubacc  6314  tfrcllemubacc  6327