ofpreima2 - Mathbox for Thierry Arnoux

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Theorem ofpreima2 32648

Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 32647 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 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘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 32647	. . 3 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
6		inundif 4426	. . . . 5 ⊢ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (^◡𝑅 “ 𝐷)
7		iuneq1 4956	. . . . 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 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
8	6, 7	ax-mp 5	. . . 4 ⊢ ∪ 𝑝 ∈ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))
9		iunxun 5040	. . . 4 ⊢ ∪ 𝑝 ∈ (((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
10	8, 9	eqtr3i 2756	. . 3 ⊢ ∪ 𝑝 ∈ (^◡𝑅 “ 𝐷)((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
11	5, 10	eqtrdi 2782	. 2 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))))
12		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)))
13	12	eldifbd 3910	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14		cnvimass 6030	. . . . . . . . . . . . . 14 ⊢ (^◡𝑅 “ 𝐷) ⊆ dom 𝑅
15	4	fndmd 6586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
16	14, 15	sseqtrid 3972	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑅 “ 𝐷) ⊆ (𝐵 × 𝐶))
17	16	ssdifssd 4094	. . . . . . . . . . . 12 ⊢ (𝜑 → ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
18	17	sselda 3929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
19		1st2nd2 7960	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
20		elxp6 7955	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∧ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺)))
21	20	simplbi2 500	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ → (((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
22	18, 19, 21	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
23	13, 22	mtod 198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺))
24		ianor 983	. . . . . . . . 9 ⊢ (¬ ((1^st ‘𝑝) ∈ ran 𝐹 ∧ (2^nd ‘𝑝) ∈ ran 𝐺) ↔ (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
25	23, 24	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
26		disjsn 4661	. . . . . . . . 9 ⊢ ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ↔ ¬ (1^st ‘𝑝) ∈ ran 𝐹)
27		disjsn 4661	. . . . . . . . 9 ⊢ ((ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅ ↔ ¬ (2^nd ‘𝑝) ∈ ran 𝐺)
28	26, 27	orbi12i 914	. . . . . . . 8 ⊢ (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) ↔ (¬ (1^st ‘𝑝) ∈ ran 𝐹 ∨ ¬ (2^nd ‘𝑝) ∈ ran 𝐺))
29	25, 28	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅))
30	1	ffnd 6652	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴)
31		dffn3 6663	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
32	30, 31	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
33	2	ffnd 6652	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐴)
34		dffn3 6663	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺:𝐴⟶ran 𝐺)
35	33, 34	sylib 218	. . . . . . . . 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 6701	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅) → (^◡𝐹 “ {(1^st ‘𝑝)}) = ∅)
38		ineq1 4160	. . . . . . . . . . . 12 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = (∅ ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
39		0in 4344	. . . . . . . . . . . 12 ⊢ (∅ ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅
40	38, 39	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
41	37, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
42	41	ex 412	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
43		fimacnvdisj 6701	. . . . . . . . . . 11 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → (^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅)
44		ineq2 4161	. . . . . . . . . . . 12 ⊢ ((^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ ∅))
45		in0 4342	. . . . . . . . . . . 12 ⊢ ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ ∅) = ∅
46	44, 45	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ((^◡𝐺 “ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
47	43, 46	syl 17	. . . . . . . . . 10 ⊢ ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
48	47	ex 412	. . . . . . . . 9 ⊢ (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅ → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
49	42, 48	jaao 956	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ 𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
50	32, 36, 49	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1^st ‘𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2^nd ‘𝑝)}) = ∅) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅))
51	29, 50	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
52	51	iuneq2dv 4964	. . . . 5 ⊢ (𝜑 → ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
53		iun0 5008	. . . . 5 ⊢ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
54	52, 53	eqtrdi 2782	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) = ∅)
55	54	uneq2d 4115	. . 3 ⊢ (𝜑 → (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))) = (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∅))
56		un0 4341	. . 3 ⊢ (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∅) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))
57	55, 56	eqtrdi 2782	. 2 ⊢ (𝜑 → (∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})) ∪ ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∖ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)}))) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))
58	11, 57	eqtrd 2766	1 ⊢ (𝜑 → (^◡(𝐹 ∘_f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((^◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((^◡𝐹 “ {(1^st ‘𝑝)}) ∩ (^◡𝐺 “ {(2^nd ‘𝑝)})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ∅c0 4280 {csn 4573 ⟨cop 4579 ∪ ciun 4939 × cxp 5612 ^◡ccnv 5613 dom cdm 5614 ran crn 5615 “ cima 5617 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 1^st c1st 7919 2^nd c2nd 7920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-1st 7921 df-2nd 7922

This theorem is referenced by: sibfof 34353

W3C validator