Theorem ofpreima2 32446

Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 32445 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 32445	. . 3 ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
6		inundif 4475	. . . . 5 ⊢ (((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (◡𝑅 “ 𝐷)
7		iuneq1 5008	. . . . 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 5092	. . . 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 2758	. . 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	eqtrdi 2784	. 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))
13	12	eldifbd 3958	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14		cnvimass 6080	. . . . . . . . . . . . . 14 ⊢ (◡𝑅 “ 𝐷) ⊆ dom 𝑅
15	4	fndmd 6654	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
16	14, 15	sseqtrid 4031	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
17	16	ssdifssd 4139	. . . . . . . . . . . 12 ⊢ (𝜑 → ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
18	17	sselda 3979	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
19		1st2nd2 8027	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
20		elxp6 8022	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ ∧ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺)))
21	20	simplbi2 500	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ → (((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
22	18, 19, 21	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
23	13, 22	mtod 197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺))
24		ianor 980	. . . . . . . . 9 ⊢ (¬ ((1st ‘𝑝) ∈ ran 𝐹 ∧ (2nd ‘𝑝) ∈ ran 𝐺) ↔ (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
25	23, 24	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
26		disjsn 4712	. . . . . . . . 9 ⊢ ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ↔ ¬ (1st ‘𝑝) ∈ ran 𝐹)
27		disjsn 4712	. . . . . . . . 9 ⊢ ((ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅ ↔ ¬ (2nd ‘𝑝) ∈ ran 𝐺)
28	26, 27	orbi12i 913	. . . . . . . 8 ⊢ (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) ↔ (¬ (1st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2nd ‘𝑝) ∈ ran 𝐺))
29	25, 28	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅))
30	1	ffnd 6718	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
31		dffn3 6730	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
32	30, 31	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
33	2	ffnd 6718	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
34		dffn3 6730	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺:𝐴⟶ran 𝐺)
35	33, 34	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐴⟶ran 𝐺)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
37		fimacnvdisj 6770	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅) → (◡𝐹 “ {(1st ‘𝑝)}) = ∅)
38		ineq1 4202	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = (∅ ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
39		0in 4390	. . . . . . . . . . . 12 ⊢ (∅ ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅
40	38, 39	eqtrdi 2784	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
41	37, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
42	41	ex 412	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
43		fimacnvdisj 6770	. . . . . . . . . . 11 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → (◡𝐺 “ {(2nd ‘𝑝)}) = ∅)
44		ineq2 4203	. . . . . . . . . . . 12 ⊢ ((◡𝐺 “ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ((◡𝐹 “ {(1st ‘𝑝)}) ∩ ∅))
45		in0 4388	. . . . . . . . . . . 12 ⊢ ((◡𝐹 “ {(1st ‘𝑝)}) ∩ ∅) = ∅
46	44, 45	eqtrdi 2784	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
47	43, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
48	47	ex 412	. . . . . . . . 9 ⊢ (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅ → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
49	42, 48	jaao 953	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
50	32, 36, 49	syl2an2r 684	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd ‘𝑝)}) = ∅) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅))
51	29, 50	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
52	51	iuneq2dv 5016	. . . . 5 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
53		iun0 5060	. . . . 5 ⊢ ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
54	52, 53	eqtrdi 2784	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})) = ∅)
55	54	uneq2d 4160	. . 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)})) ∪ ∅))
56		un0 4387	. . 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)}))
57	55, 56	eqtrdi 2784	. 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)})))
58	11, 57	eqtrd 2768	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   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ∖ cdif 3942   ∪ cun 3943   ∩ cin 3944  ∅c0 4319  {csn 4625  ⟨cop 4631  ∪ ciun 4992   × cxp 5671  ^◡ccnv 5672  dom cdm 5673  ran crn 5674   “ cima 5676   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7415   ∘_f cof 7678  1^st c1st 7986  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7680  df-1st 7988  df-2nd 7989

This theorem is referenced by:  sibfof  33955