Theorem ofpreima 32480

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 5260	. . . . . . 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 2729	. . . . . . 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 7559	. . . . . . . 8 ⊢ (𝑅 Fn (𝐵 × 𝐶) ↔ 𝑅 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
8	6, 7	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
9	1, 2, 3, 4, 5, 8	ofoprabco 32479	. . . . . 6 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)))
10	9	cnveqd 5882	. . . . 5 ⊢ (𝜑 → ◡(𝐹 ∘f 𝑅𝐺) = ◡(𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)))
11		cnvco 5892	. . . . 5 ⊢ ◡(𝑅 ∘ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅)
12	10, 11	eqtrdi 2784	. . . 4 ⊢ (𝜑 → ◡(𝐹 ∘f 𝑅𝐺) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅))
13	12	imaeq1d 6067	. . 3 ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ((◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅) “ 𝐷))
14		imaco 6260	. . 3 ⊢ ((◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ∘ ◡𝑅) “ 𝐷) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷))
15	13, 14	eqtrdi 2784	. 2 ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)))
16		dfima2 6070	. . 3 ⊢ (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)) = {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞}
17		vex 3477	. . . . . . . 8 ⊢ 𝑝 ∈ V
18		vex 3477	. . . . . . . 8 ⊢ 𝑞 ∈ V
19	17, 18	brcnv 5889	. . . . . . 7 ⊢ (𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞 ↔ 𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝)
20		funmpt 6596	. . . . . . . . 9 ⊢ Fun (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)
21		funbrfv2b 6961	. . . . . . . . 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 5470	. . . . . . . . . . 11 ⊢ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩ ∈ V
24		eqid 2728	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) = (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)
25	23, 24	dmmpti 6704	. . . . . . . . . 10 ⊢ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) = 𝐴
26	25	eleq2i 2821	. . . . . . . . 9 ⊢ (𝑞 ∈ dom (𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) ↔ 𝑞 ∈ 𝐴)
27	26	anbi1i 622	. . . . . . . 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 274	. . . . . . 7 ⊢ (𝑞(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑝 ↔ (𝑞 ∈ 𝐴 ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝))
29		fveq2 6902	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑞 → (𝐹‘𝑠) = (𝐹‘𝑞))
30		fveq2 6902	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑞 → (𝐺‘𝑠) = (𝐺‘𝑞))
31	29, 30	opeq12d 4886	. . . . . . . . . 10 ⊢ (𝑠 = 𝑞 → ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩ = ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩)
32		opex 5470	. . . . . . . . . 10 ⊢ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ ∈ V
33	31, 24, 32	fvmpt 7010	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐴 → ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩)
34	33	eqeq1d 2730	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
35	34	pm5.32i 573	. . . . . . 7 ⊢ ((𝑞 ∈ 𝐴 ∧ ((𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)‘𝑞) = 𝑝) ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
36	19, 28, 35	3bitri 296	. . . . . 6 ⊢ (𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞 ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
37	36	rexbii 3091	. . . . 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 2798	. . . 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 1909	. . . . 5 ⊢ Ⅎ𝑞𝜑
40		nfab1 2901	. . . . 5 ⊢ Ⅎ𝑞{𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)}
41		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑞∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))
42		eliun 5004	. . . . . 6 ⊢ (𝑞 ∈ ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
43		ffn 6727	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
44		fniniseg 7074	. . . . . . . . . . . . 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 6727	. . . . . . . . . . . . 13 ⊢ (𝐺:𝐴⟶𝐶 → 𝐺 Fn 𝐴)
47		fniniseg 7074	. . . . . . . . . . . . 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 630	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ∧ 𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ((𝑞 ∈ 𝐴 ∧ (𝐹‘𝑞) = (1st ‘𝑝)) ∧ (𝑞 ∈ 𝐴 ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
50		elin 3965	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ (◡𝐹 “ {(1st ‘𝑝)}) ∧ 𝑞 ∈ (◡𝐺 “ {(2nd ‘𝑝)})))
51		anandi 674	. . . . . . . . . . 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 313	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
53	52	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
54		cnvimass 6090	. . . . . . . . . . . . . 14 ⊢ (◡𝑅 “ 𝐷) ⊆ dom 𝑅
55	6	fndmd 6664	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
56	54, 55	sseqtrid 4034	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
57	56	sselda 3982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → 𝑝 ∈ (𝐵 × 𝐶))
58		1st2nd2 8040	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
59		eqeq2 2740	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
60	57, 58, 59	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩))
61		fvex 6915	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑞) ∈ V
62		fvex 6915	. . . . . . . . . . . 12 ⊢ (𝐺‘𝑞) ∈ V
63	61, 62	opth 5482	. . . . . . . . . . 11 ⊢ (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ↔ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))
64	60, 63	bitrdi 286	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝 ↔ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝))))
65	64	anbi2d 628	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → ((𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝) ↔ (𝑞 ∈ 𝐴 ∧ ((𝐹‘𝑞) = (1st ‘𝑝) ∧ (𝐺‘𝑞) = (2nd ‘𝑝)))))
66	53, 65	bitr4d 281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (◡𝑅 “ 𝐷)) → (𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ (𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)))
67	66	rexbidva 3174	. . . . . . 7 ⊢ (𝜑 → (∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)))
68		abid 2709	. . . . . . 7 ⊢ (𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} ↔ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝))
69	67, 68	bitr4di 288	. . . . . 6 ⊢ (𝜑 → (∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑞 ∈ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ↔ 𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)}))
70	42, 69	bitr2id 283	. . . . 5 ⊢ (𝜑 → (𝑞 ∈ {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} ↔ 𝑞 ∈ ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
71	39, 40, 41, 70	eqrd 4001	. . . 4 ⊢ (𝜑 → {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)(𝑞 ∈ 𝐴 ∧ ⟨(𝐹‘𝑞), (𝐺‘𝑞)⟩ = 𝑝)} = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
72	38, 71	eqtrid 2780	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑝 ∈ (◡𝑅 “ 𝐷)𝑝◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩)𝑞} = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
73	16, 72	eqtrid 2780	. 2 ⊢ (𝜑 → (◡(𝑠 ∈ 𝐴 ↦ ⟨(𝐹‘𝑠), (𝐺‘𝑠)⟩) “ (◡𝑅 “ 𝐷)) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
74	15, 73	eqtrd 2768	1 ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  ∃wrex 3067   ∩ cin 3948  {csn 4632  ⟨cop 4638  ∪ ciun 5000   class class class wbr 5152   ↦ cmpt 5235   × cxp 5680  ^◡ccnv 5681  dom cdm 5682   “ cima 5685   ∘ ccom 5686  Fun wfun 6547   Fn wfn 6548  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428   ∘_f cof 7690  1^st c1st 7999  2^nd c2nd 8000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7692  df-1st 8001  df-2nd 8002

This theorem is referenced by:  ofpreima2  32481