Theorem fun2ssres 3553

Description: Equality of restrictions of a function and a subclass.

Assertion

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

Proof of Theorem fun2ssres

Step	Hyp	Ref	Expression
1		resabs1 3388	. . . 4 |- (A (_ dom G -> ((F |` dom G) |` A) = (F |` A))
2	1	eqcomd 1480	. . 3 |- (A (_ dom G -> (F |` A) = ((F |` dom G) |` A))
3		funssres 3552	. . . 4 |- ((Fun F /\ G (_ F) -> (F |` dom G) = G)
4		reseq1 3368	. . . 4 |- ((F |` dom G) = G -> ((F |` dom G) |` A) = (G |` A))
5	3, 4	syl 10	. . 3 |- ((Fun F /\ G (_ F) -> ((F |` dom G) |` A) = (G |` A))
6	2, 5	sylan9eqr 1529	. 2 |- (((Fun F /\ G (_ F) /\ A (_ dom G) -> (F |` A) = (G |` A))
7	6	3impa 828	1 |- ((Fun F /\ G (_ F /\ A (_ dom G) -> (F |` A) = (G |` A))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   (_ wss 2047  dom cdm 3170   |` cres 3172  Fun wfun 3176

This theorem is referenced by:  tfrlem9 3919  tfrlem11 3921  subgres 8117

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-res 3190  df-fun 3192

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