Theorem xppreima 33027

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)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍)))

Proof of Theorem xppreima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6573	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fncnvima2 7063	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
3	1, 2	sylbi 220	. . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
4	3	adantr 486	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
5		elxp6 8029	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩ ∧ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
6		fvco 6986	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
7		fvco 6986	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
8	6, 7	opeq12d 4851	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
9	8	eqeq2d 2777	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ↔ (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
10	6	eleq1d 2851	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ (1st ‘(𝐹‘𝑥)) ∈ 𝑌))
11	7	eleq1d 2851	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍))
12	10, 11	anbi12d 644	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
13	9, 12	anbi12d 644	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍)) ↔ ((𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩ ∧ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍))))
14	5, 13	bitr4id 293	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
15	14	adantlr 728	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
16		opfv 33026	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)
17	16	biantrurd 542	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ ((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ∧ (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍))))
18		fo1st 8015	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
19		fofun 6800	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → Fun 1st )
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 1st
21		funco 6583	. . . . . . . . . 10 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
22	20, 21	mpan 703	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (1st ∘ 𝐹))
23	22	adantr 486	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (1st ∘ 𝐹))
24		ssv 3964	. . . . . . . . . . . 12 ⊢ (𝐹 “ dom 𝐹) ⊆ V
25		fof 6799	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st :V⟶V)
26		fdm 6722	. . . . . . . . . . . . 13 ⊢ (1st :V⟶V → dom 1st = V)
27	18, 25, 26	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 1st = V
28	24, 27	sseqtrri 3989	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 1st
29		ssid 3962	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ dom 𝐹
30		funimass3 7056	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
31	29, 30	mpan2 704	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
32	28, 31	mpbii 236	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 1st ))
33	32	sselda 3940	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 1st ))
34		dmco 6261	. . . . . . . . 9 ⊢ dom (1st ∘ 𝐹) = (◡𝐹 “ dom 1st )
35	33, 34	eleqtrrdi 2877	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1st ∘ 𝐹))
36		fvimacnv 7055	. . . . . . . 8 ⊢ ((Fun (1st ∘ 𝐹) ∧ 𝑥 ∈ dom (1st ∘ 𝐹)) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
37	23, 35, 36	syl2anc 596	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
38		fo2nd 8016	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
39		fofun 6800	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → Fun 2nd )
40	38, 39	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 2nd
41		funco 6583	. . . . . . . . . 10 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
42	40, 41	mpan 703	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (2nd ∘ 𝐹))
43	42	adantr 486	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (2nd ∘ 𝐹))
44		fof 6799	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
45		fdm 6722	. . . . . . . . . . . . 13 ⊢ (2nd :V⟶V → dom 2nd = V)
46	38, 44, 45	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 2nd = V
47	24, 46	sseqtrri 3989	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 2nd
48		funimass3 7056	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
49	29, 48	mpan2 704	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
50	47, 49	mpbii 236	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 2nd ))
51	50	sselda 3940	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 2nd ))
52		dmco 6261	. . . . . . . . 9 ⊢ dom (2nd ∘ 𝐹) = (◡𝐹 “ dom 2nd )
53	51, 52	eleqtrrdi 2877	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2nd ∘ 𝐹))
54		fvimacnv 7055	. . . . . . . 8 ⊢ ((Fun (2nd ∘ 𝐹) ∧ 𝑥 ∈ dom (2nd ∘ 𝐹)) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
55	43, 53, 54	syl2anc 596	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
56	37, 55	anbi12d 644	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))))
57	56	adantlr 728	. . . . 5 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))))
58	15, 17, 57	3bitr2d 310	. . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))))
59	58	rabbidva 3425	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)} = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))})
60	4, 59	eqtrd 2801	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))})
61		dfin5 3916	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)}
62		dfin5 3916	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)}
63	61, 62	ineq12i 4174	. . 3 ⊢ ((dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍))) = ({𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)})
64		cnvimass 6089	. . . . . 6 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom (1st ∘ 𝐹)
65		dmcoss 5970	. . . . . 6 ⊢ dom (1st ∘ 𝐹) ⊆ dom 𝐹
66	64, 65	sstri 3949	. . . . 5 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹
67		sseqin2 4179	. . . . 5 ⊢ ((◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌))
68	66, 67	mpbi 233	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌)
69		cnvimass 6089	. . . . . 6 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom (2nd ∘ 𝐹)
70		dmcoss 5970	. . . . . 6 ⊢ dom (2nd ∘ 𝐹) ⊆ dom 𝐹
71	69, 70	sstri 3949	. . . . 5 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹
72		sseqin2 4179	. . . . 5 ⊢ ((◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍))
73	71, 72	mpbi 233	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍)
74	68, 73	ineq12i 4174	. . 3 ⊢ ((dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍))) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
75		inrab 4272	. . 3 ⊢ ({𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)}) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))}
76	63, 74, 75	3eqtr3ri 2798	. 2 ⊢ {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))} = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
77	60, 76	eqtrdi 2817	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  ⟨cop 4600   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   “ cima 5669   ∘ ccom 5670  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  1^st c1st 7993  2^nd c2nd 7994

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7995  df-2nd 7996

This theorem is used by:  xppreima2  33033