Theorem oprssoprval 4034

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

Assertion

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

Proof of Theorem oprssoprval

Step	Hyp	Ref	Expression
1		oprvalres 4033	. . 3 |- ((A e. C /\ B e. D) -> (A(F |` (C X. D))B) = (AFB))
2	1	adantl 388	. 2 |- (((Fun F /\ G Fn (C X. D) /\ G (_ F) /\ (A e. C /\ B e. D)) -> (A(F |` (C X. D))B) = (AFB))
3		fndm 3587	. . . . . . 7 |- (G Fn (C X. D) -> dom G = (C X. D))
4		reseq2 3369	. . . . . . 7 |- (dom G = (C X. D) -> (F |` dom G) = (F |` (C X. D)))
5	3, 4	syl 10	. . . . . 6 |- (G Fn (C X. D) -> (F |` dom G) = (F |` (C X. D)))
6	5	3ad2ant2 801	. . . . 5 |- ((Fun F /\ G Fn (C X. D) /\ G (_ F) -> (F |` dom G) = (F |` (C X. D)))
7		funssres 3552	. . . . . 6 |- ((Fun F /\ G (_ F) -> (F |` dom G) = G)
8	7	3adant2 798	. . . . 5 |- ((Fun F /\ G Fn (C X. D) /\ G (_ F) -> (F |` dom G) = G)
9	6, 8	eqtr3d 1509	. . . 4 |- ((Fun F /\ G Fn (C X. D) /\ G (_ F) -> (F |` (C X. D)) = G)
10	9	opreqd 3977	. . 3 |- ((Fun F /\ G Fn (C X. D) /\ G (_ F) -> (A(F |` (C X. D))B) = (AGB))
11	10	adantr 389	. 2 |- (((Fun F /\ G Fn (C X. D) /\ G (_ F) /\ (A e. C /\ B e. D)) -> (A(F |` (C X. D))B) = (AGB))
12	2, 11	eqtr3d 1509	1 |- (((Fun F /\ G Fn (C X. D) /\ G (_ F) /\ (A e. C /\ B e. D)) -> (AFB) = (AGB))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   (_ wss 2047   X. cxp 3168  dom cdm 3170   |` cres 3172  Fun wfun 3176   Fn wfn 3177  (class class class)co 3963

This theorem is referenced by:  sspg 8387  ssps 8389

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-uni 2504  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-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965

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