Theorem ofpreima 30186

Description: Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)

Hypotheses

Ref	Expression
ofpreima.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofpreima.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
ofpreima.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofpreima.4	⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))

Assertion

Ref	Expression
ofpreima	⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))

Distinct variable groups:   𝐴,𝑝   𝐷,𝑝   𝐹,𝑝   𝐺,𝑝   𝑅,𝑝   𝜑,𝑝

Allowed substitution hints:   𝐵(𝑝)   𝐶(𝑝)   𝑉(𝑝)

Proof of Theorem ofpreima

Dummy variables 𝑞 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfmpt1 5021	. . . . . . 7 ⊢ Ⅎ𝑠(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)
2		ofpreima.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3		ofpreima.2	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
4		ofpreima.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		eqidd 2773	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) = (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩))
6		ofpreima.4	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))
7		fnov 7096	. . . . . . . 8 ⊢ (𝑅 Fn (𝐵 × 𝐶) ↔ 𝑅 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
8	6, 7	sylib 210	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
9	1, 2, 3, 4, 5, 8	ofoprabco 30185	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)))
10	9	cnveqd 5592	. . . . 5 ⊢ (𝜑 → ◡(𝐹 ∘𝑓 𝑅𝐺) = ◡(𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)))
11		cnvco 5602	. . . . 5 ⊢ ◡(𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅)
12	10, 11	syl6eq 2824	. . . 4 ⊢ (𝜑 → ◡(𝐹 ∘𝑓 𝑅𝐺) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅))
13	12	imaeq1d 5766	. . 3 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ((◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅) “ 𝐷))
14		imaco 5940	. . 3 ⊢ ((◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅) “ 𝐷) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷))
15	13, 14	syl6eq 2824	. 2 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)))
16		dfima2 5769	. . 3 ⊢ (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)) = {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞}
17		vex 3412	. . . . . . . 8 ⊢ 𝑝 ∈ V
18		vex 3412	. . . . . . . 8 ⊢ 𝑞 ∈ V
19	17, 18	brcnv 5599	. . . . . . 7 ⊢ (𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞 ↔ 𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝)
20		funmpt 6223	. . . . . . . . 9 ⊢ Fun (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)
21		funbrfv2b 6550	. . . . . . . . 9 ⊢ (Fun (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) → (𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝 ↔ (𝑞 ∈ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝)))
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ (𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝 ↔ (𝑞 ∈ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝))
23		opex 5209	. . . . . . . . . . 11 ⊢ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩ ∈ V
24		eqid 2772	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) = (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)
25	23, 24	dmmpti 6319	. . . . . . . . . 10 ⊢ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) = 𝐴
26	25	eleq2i 2851	. . . . . . . . 9 ⊢ (𝑞 ∈ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ↔ 𝑞 ∈ 𝐴)
27	26	anbi1i 614	. . . . . . . 8 ⊢ ((𝑞 ∈ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝) ↔ (𝑞 ∈ 𝐴 ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝))
28	22, 27	bitri 267	. . . . . . 7 ⊢ (𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝 ↔ (𝑞 ∈ 𝐴 ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝))
29		fveq2 6496	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑞 → (𝐹‘𝑠) = (𝐹‘𝑞))
30		fveq2 6496	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑞 → (𝐺‘𝑠) = (𝐺‘𝑞))
31	29, 30	opeq12d 4681	. . . . . . . . . 10 ⊢ (𝑠 = 𝑞 → ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩ = ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩)
32		opex 5209	. . . . . . . . . 10 ⊢ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ ∈ V
33	31, 24, 32	fvmpt 6593	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩)
34	33	eqeq1d 2774	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
35	34	pm5.32i 567	. . . . . . 7 ⊢ ((𝑞 ∈ 𝐴 ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝) ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
36	19, 28, 35	3bitri 289	. . . . . 6 ⊢ (𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞 ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
37	36	rexbii 3188	. . . . 5 ⊢ (∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞 ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
38	37	abbii 2838	. . . 4 ⊢ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞} = {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)}
39		nfv 1873	. . . . 5 ⊢ Ⅎ𝑞𝜑
40		nfab1 2928	. . . . 5 ⊢ Ⅎ𝑞{𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)}
41		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑞∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))
42		eliun 4792	. . . . . 6 ⊢ (𝑞 ∈ ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
43		ffn 6341	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
44		fniniseg 6653	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ↔ (𝑞 ∈ 𝐴 ∧ (𝐹‘𝑞) = (1st ‘𝑝))))
45	2, 43, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ↔ (𝑞 ∈ 𝐴 ∧ (𝐹‘𝑞) = (1st ‘𝑝))))
46		ffn 6341	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶𝐶 → 𝐺 Fn 𝐴)
47		fniniseg 6653	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝐴 → (𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)}) ↔ (𝑞 ∈ 𝐴 ∧ (𝐺‘𝑞) = (2nd ‘𝑝))))
48	3, 46, 47	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)}) ↔ (𝑞 ∈ 𝐴 ∧ (𝐺‘𝑞) = (2nd ‘𝑝))))
49	45, 48	anbi12d 621	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ∧ 𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ((𝑞 ∈ 𝐴 ∧ (𝐹‘𝑞) = (1st ‘𝑝)) ∧ (𝑞 ∈ 𝐴 ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
50		elin 4051	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ∧ 𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)})))
51		anandi 663	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝))) ↔ ((𝑞 ∈ 𝐴 ∧ (𝐹‘𝑞) = (1st ‘𝑝)) ∧ (𝑞 ∈ 𝐴 ∧ (𝐺‘𝑞) = (2nd ‘𝑝))))
52	49, 50, 51	3bitr4g 306	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
53	52	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
54		cnvimass 5786	. . . . . . . . . . . . . 14 ⊢ (◡𝑅 “ 𝐷) ⊆ dom 𝑅
55		fndm 6285	. . . . . . . . . . . . . . 15 ⊢ (𝑅 Fn (𝐵 × 𝐶) → dom 𝑅 = (𝐵 × 𝐶))
56	6, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
57	54, 56	syl5sseq 3903	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
58	57	sselda 3852	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → 𝑝 ∈ (𝐵 × 𝐶))
59		1st2nd2 7538	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
60		eqeq2 2783	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
61	58, 59, 60	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
62		fvex 6509	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑞) ∈ V
63		fvex 6509	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑞) ∈ V
64	62, 63	opth 5221	. . . . . . . . . . 11 ⊢ (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))
65	61, 64	syl6bb 279	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝))))
66	65	anbi2d 619	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → ((𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
67	53, 66	bitr4d 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)))
68	67	rexbidva 3235	. . . . . . 7 ⊢ (𝜑 → (∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)))
69		abid 2756	. . . . . . 7 ⊢ (𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
70	68, 69	syl6bbr 281	. . . . . 6 ⊢ (𝜑 → (∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ 𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)}))
71	42, 70	syl5rbb 276	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} ↔ 𝑞 ∈ ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
72	39, 40, 41, 71	eqrd 3871	. . . 4 ⊢ (𝜑 → {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
73	38, 72	syl5eq 2820	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞} = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
74	16, 73	syl5eq 2820	. 2 ⊢ (𝜑 → (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
75	15, 74	eqtrd 2808	1 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2752  ∃wrex 3083   ∩ cin 3822  {csn 4435  ⟨cop 4441  ∪ ciun 4788   class class class wbr 4925   ↦ cmpt 5004   × cxp 5401  ^◡ccnv 5402  dom cdm 5403   “ cima 5406   ∘ ccom 5407  Fun wfun 6179   Fn wfn 6180  ⟶wf 6181  ‘cfv 6185  (class class class)co 6974   ∈ cmpo 6976   ∘_𝑓 cof 7223  1^st c1st 7497  2^nd c2nd 7498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-of 7225  df-1st 7499  df-2nd 7500

This theorem is referenced by:  ofpreima2  30187