xppreima - Mathbox for Thierry Arnoux

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Theorem xppreima 31859

Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)

**Assertion**
Ref	Expression
xppreima	⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (^◡𝐹 “ (𝑌 × 𝑍)) = ((^◡(1^st ∘ 𝐹) “ 𝑌) ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍)))

Proof of Theorem xppreima Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6576	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fncnvima2 7060	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (^◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
3	1, 2	sylbi 216	. . . 4 ⊢ (Fun 𝐹 → (^◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
4	3	adantr 482	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (^◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
5		elxp6 8006	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩ ∧ ((1^st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2^nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
6		fvco 6987	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1^st ∘ 𝐹)‘𝑥) = (1^st ‘(𝐹‘𝑥)))
7		fvco 6987	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2^nd ∘ 𝐹)‘𝑥) = (2^nd ‘(𝐹‘𝑥)))
8	6, 7	opeq12d 4881	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
9	8	eqeq2d 2744	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ ↔ (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
10	6	eleq1d 2819	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ (1^st ‘(𝐹‘𝑥)) ∈ 𝑌))
11	7	eleq1d 2819	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ (2^nd ‘(𝐹‘𝑥)) ∈ 𝑍))
12	10, 11	anbi12d 632	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ ((1^st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2^nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
13	9, 12	anbi12d 632	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍)) ↔ ((𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩ ∧ ((1^st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2^nd ‘(𝐹‘𝑥)) ∈ 𝑍))))
14	5, 13	bitr4id 290	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
15	14	adantlr 714	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
16		opfv 31858	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩)
17	16	biantrurd 534	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1^st ∘ 𝐹)‘𝑥), ((2^nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
18		fo1st 7992	. . . . . . . . . . 11 ⊢ 1^st :V–onto→V
19		fofun 6804	. . . . . . . . . . 11 ⊢ (1^st :V–onto→V → Fun 1^st )
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 1^st
21		funco 6586	. . . . . . . . . 10 ⊢ ((Fun 1^st ∧ Fun 𝐹) → Fun (1^st ∘ 𝐹))
22	20, 21	mpan 689	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (1^st ∘ 𝐹))
23	22	adantr 482	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (1^st ∘ 𝐹))
24		ssv 4006	. . . . . . . . . . . 12 ⊢ (𝐹 “ dom 𝐹) ⊆ V
25		fof 6803	. . . . . . . . . . . . 13 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
26		fdm 6724	. . . . . . . . . . . . 13 ⊢ (1^st :V⟶V → dom 1^st = V)
27	18, 25, 26	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 1^st = V
28	24, 27	sseqtrri 4019	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 1^st
29		ssid 4004	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ dom 𝐹
30		funimass3 7053	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1^st ↔ dom 𝐹 ⊆ (^◡𝐹 “ dom 1^st )))
31	29, 30	mpan2 690	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1^st ↔ dom 𝐹 ⊆ (^◡𝐹 “ dom 1^st )))
32	28, 31	mpbii 232	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (^◡𝐹 “ dom 1^st ))
33	32	sselda 3982	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (^◡𝐹 “ dom 1^st ))
34		dmco 6251	. . . . . . . . 9 ⊢ dom (1^st ∘ 𝐹) = (^◡𝐹 “ dom 1^st )
35	33, 34	eleqtrrdi 2845	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1^st ∘ 𝐹))
36		fvimacnv 7052	. . . . . . . 8 ⊢ ((Fun (1^st ∘ 𝐹) ∧ 𝑥 ∈ dom (1^st ∘ 𝐹)) → (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌)))
37	23, 35, 36	syl2anc 585	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌)))
38		fo2nd 7993	. . . . . . . . . . 11 ⊢ 2^nd :V–onto→V
39		fofun 6804	. . . . . . . . . . 11 ⊢ (2^nd :V–onto→V → Fun 2^nd )
40	38, 39	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 2^nd
41		funco 6586	. . . . . . . . . 10 ⊢ ((Fun 2^nd ∧ Fun 𝐹) → Fun (2^nd ∘ 𝐹))
42	40, 41	mpan 689	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (2^nd ∘ 𝐹))
43	42	adantr 482	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (2^nd ∘ 𝐹))
44		fof 6803	. . . . . . . . . . . . 13 ⊢ (2^nd :V–onto→V → 2^nd :V⟶V)
45		fdm 6724	. . . . . . . . . . . . 13 ⊢ (2^nd :V⟶V → dom 2^nd = V)
46	38, 44, 45	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 2^nd = V
47	24, 46	sseqtrri 4019	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 2^nd
48		funimass3 7053	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2^nd ↔ dom 𝐹 ⊆ (^◡𝐹 “ dom 2^nd )))
49	29, 48	mpan2 690	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2^nd ↔ dom 𝐹 ⊆ (^◡𝐹 “ dom 2^nd )))
50	47, 49	mpbii 232	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (^◡𝐹 “ dom 2^nd ))
51	50	sselda 3982	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (^◡𝐹 “ dom 2^nd ))
52		dmco 6251	. . . . . . . . 9 ⊢ dom (2^nd ∘ 𝐹) = (^◡𝐹 “ dom 2^nd )
53	51, 52	eleqtrrdi 2845	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2^nd ∘ 𝐹))
54		fvimacnv 7052	. . . . . . . 8 ⊢ ((Fun (2^nd ∘ 𝐹) ∧ 𝑥 ∈ dom (2^nd ∘ 𝐹)) → (((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍)))
55	43, 53, 54	syl2anc 585	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍)))
56	37, 55	anbi12d 632	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))))
57	56	adantlr 714	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1^st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2^nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))))
58	15, 17, 57	3bitr2d 307	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))))
59	58	rabbidva 3440	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)} = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))})
60	4, 59	eqtrd 2773	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (^◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))})
61		dfin5 3956	. . . 4 ⊢ (dom 𝐹 ∩ (^◡(1^st ∘ 𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌)}
62		dfin5 3956	. . . 4 ⊢ (dom 𝐹 ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍)}
63	61, 62	ineq12i 4210	. . 3 ⊢ ((dom 𝐹 ∩ (^◡(1^st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍))) = ({𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍)})
64		cnvimass 6078	. . . . . 6 ⊢ (^◡(1^st ∘ 𝐹) “ 𝑌) ⊆ dom (1^st ∘ 𝐹)
65		dmcoss 5969	. . . . . 6 ⊢ dom (1^st ∘ 𝐹) ⊆ dom 𝐹
66	64, 65	sstri 3991	. . . . 5 ⊢ (^◡(1^st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹
67		sseqin2 4215	. . . . 5 ⊢ ((^◡(1^st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (^◡(1^st ∘ 𝐹) “ 𝑌)) = (^◡(1^st ∘ 𝐹) “ 𝑌))
68	66, 67	mpbi 229	. . . 4 ⊢ (dom 𝐹 ∩ (^◡(1^st ∘ 𝐹) “ 𝑌)) = (^◡(1^st ∘ 𝐹) “ 𝑌)
69		cnvimass 6078	. . . . . 6 ⊢ (^◡(2^nd ∘ 𝐹) “ 𝑍) ⊆ dom (2^nd ∘ 𝐹)
70		dmcoss 5969	. . . . . 6 ⊢ dom (2^nd ∘ 𝐹) ⊆ dom 𝐹
71	69, 70	sstri 3991	. . . . 5 ⊢ (^◡(2^nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹
72		sseqin2 4215	. . . . 5 ⊢ ((^◡(2^nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍)) = (^◡(2^nd ∘ 𝐹) “ 𝑍))
73	71, 72	mpbi 229	. . . 4 ⊢ (dom 𝐹 ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍)) = (^◡(2^nd ∘ 𝐹) “ 𝑍)
74	68, 73	ineq12i 4210	. . 3 ⊢ ((dom 𝐹 ∩ (^◡(1^st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍))) = ((^◡(1^st ∘ 𝐹) “ 𝑌) ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍))
75		inrab 4306	. . 3 ⊢ ({𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍)}) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))}
76	63, 74, 75	3eqtr3ri 2770	. 2 ⊢ {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (^◡(1^st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (^◡(2^nd ∘ 𝐹) “ 𝑍))} = ((^◡(1^st ∘ 𝐹) “ 𝑌) ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍))
77	60, 76	eqtrdi 2789	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (^◡𝐹 “ (𝑌 × 𝑍)) = ((^◡(1^st ∘ 𝐹) “ 𝑌) ∩ (^◡(2^nd ∘ 𝐹) “ 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 ⟨cop 4634 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 ∘ ccom 5680 Fun wfun 6535 Fn wfn 6536 ⟶wf 6537 –onto→wfo 6539 ‘cfv 6541 1^st c1st 7970 2^nd c2nd 7971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fo 6547 df-fv 6549 df-1st 7972 df-2nd 7973

This theorem is referenced by: xppreima2 31864

W3C validator