Theorem ofpreima2 28849

Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 28848 iterates the union over a smaller set. (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
ofpreima2	⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡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 ofpreima2

Step	Hyp	Ref	Expression
1		ofpreima.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		ofpreima.2	. . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
3		ofpreima.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofpreima.4	. . . 4 ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))
5	1, 2, 3, 4	ofpreima 28848	. . 3 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
6		inundif 3998	. . . . 5 ⊢ (((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (◡𝑅 “ 𝐷)
7		iuneq1 4470	. . . . 5 ⊢ ((((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (◡𝑅 “ 𝐷) → ∪ 𝑝 ∈ (((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑝 ∈ (((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))
9		iunxun 4541	. . . 4 ⊢ ∪ 𝑝 ∈ (((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
10	8, 9	eqtr3i 2634	. . 3 ⊢ ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
11	5, 10	syl6eq 2660	. 2 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))))
12		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))
13	12	eldifbd 3553	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14		cnvimass 5404	. . . . . . . . . . . . . 14 ⊢ (◡𝑅 “ 𝐷) ⊆ dom 𝑅
15		fndm 5904	. . . . . . . . . . . . . . 15 ⊢ (𝑅 Fn (𝐵 × 𝐶) → dom 𝑅 = (𝐵 × 𝐶))
16	4, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
17	14, 16	syl5sseq 3616	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
18	17	ssdifssd 3710	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
19	18	sselda 3568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
20		1st2nd2 7096	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
21		elxp6 7091	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∧ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺)))
22	21	simplbi2 653	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ → (((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
23	19, 20, 22	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
24	13, 23	mtod 188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺))
25		ianor 508	. . . . . . . . 9 ⊢ (¬ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) ↔ (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
26	24, 25	sylib 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
27		disjsn 4192	. . . . . . . . 9 ⊢ ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ↔ ¬ (1st ‘𝑝) ∈ ran 𝐹)
28		disjsn 4192	. . . . . . . . 9 ⊢ ((ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅ ↔ ¬ (2nd ‘𝑝) ∈ ran 𝐺)
29	27, 28	orbi12i 542	. . . . . . . 8 ⊢ (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) ↔ (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
30	26, 29	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅))
31	1	ffnd 5959	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
32		dffn3 5967	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
33	31, 32	sylib 207	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
34	2	ffnd 5959	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
35		dffn3 5967	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺:𝐴⟶ran 𝐺)
36	34, 35	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐴⟶ran 𝐺)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
38		fimacnvdisj 5996	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅) → (◡𝐹 “ {(1st ‘𝑝)}) = ∅)
39		ineq1 3769	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = (∅ ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
40		0in 3921	. . . . . . . . . . . 12 ⊢ (∅ ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅
41	39, 40	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
42	38, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
43	42	ex 449	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
44		fimacnvdisj 5996	. . . . . . . . . . 11 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → (◡𝐺 “ {(2nd ‘𝑝)}) = ∅)
45		ineq2 3770	. . . . . . . . . . . 12 ⊢ ((◡𝐺 “ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ((◡𝐹 “ {(1st ‘𝑝)}) ∩ ∅))
46		in0 3920	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ ∅) = ∅
47	45, 46	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
48	44, 47	syl 17	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
49	48	ex 449	. . . . . . . . 9 ⊢ (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
50	43, 49	jaao 530	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
51	33, 37, 50	syl2an2r 872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
52	30, 51	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
53	52	iuneq2dv 4478	. . . . 5 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
54		iun0 4512	. . . . 5 ⊢ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
55	53, 54	syl6eq 2660	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
56	55	uneq2d 3729	. . 3 ⊢ (𝜑 → (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) = (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∅))
57		un0 3919	. . 3 ⊢ (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∅) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))
58	56, 57	syl6eq 2660	. 2 ⊢ (𝜑 → (∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)}))) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
59	11, 58	eqtrd 2644	1 ⊢ (𝜑 → (◡(𝐹 ∘𝑓 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-1st 7059  df-2nd 7060

This theorem is referenced by:  sibfof  29729