Theorem oprssoprv 4964

Description: The value of a member of the domain of a subclass of an operation.

Assertion

Ref	Expression
oprssoprv	|- (((Fun F /\ G Fn (C X. D) /\ G C_ F) /\ (A e. C /\ B e. D)) -> (AFB) = (AGB))

Proof of Theorem oprssoprv

Step	Hyp	Ref	Expression
1		oprvres 4963	. . 3 |- ((A e. C /\ B e. D) -> (A(F |` (C X. D))B) = (AFB))
2	1	adantl 424	. 2 |- (((Fun F /\ G Fn (C X. D) /\ G C_ F) /\ (A e. C /\ B e. D)) -> (A(F |` (C X. D))B) = (AFB))
3		fndm 4512	. . . . . . 7 |- (G Fn (C X. D) -> dom G = (C X. D))
4		reseq2 4219	. . . . . . 7 |- (dom G = (C X. D) -> (F |` dom G) = (F |` (C X. D)))
5	3, 4	syl 12	. . . . . 6 |- (G Fn (C X. D) -> (F |` dom G) = (F |` (C X. D)))
6	5	3ad2ant2 898	. . . . 5 |- ((Fun F /\ G Fn (C X. D) /\ G C_ F) -> (F |` dom G) = (F |` (C X. D)))
7		funssres 4460	. . . . . 6 |- ((Fun F /\ G C_ F) -> (F |` dom G) = G)
8	7	3adant2 895	. . . . 5 |- ((Fun F /\ G Fn (C X. D) /\ G C_ F) -> (F |` dom G) = G)
9	6, 8	eqtr3d 1927	. . . 4 |- ((Fun F /\ G Fn (C X. D) /\ G C_ F) -> (F |` (C X. D)) = G)
10	9	opreqd 4899	. . 3 |- ((Fun F /\ G Fn (C X. D) /\ G C_ F) -> (A(F |` (C X. D))B) = (AGB))
11	10	adantr 425	. 2 |- (((Fun F /\ G Fn (C X. D) /\ G C_ F) /\ (A e. C /\ B e. D)) -> (A(F |` (C X. D))B) = (AGB))
12	2, 11	eqtr3d 1927	1 |- (((Fun F /\ G Fn (C X. D) /\ G C_ F) /\ (A e. C /\ B e. D)) -> (AFB) = (AGB))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593   X. cxp 3984  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  (class class class)co 4884

This theorem is referenced by:  sspg 9726  ssps 9728  subtopmetlem 10255  subtopmet 10256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886

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