Theorem xppreima 32723

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 6522	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fncnvima2 7006	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
3	1, 2	sylbi 217	. . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
4	3	adantr 480	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
5		elxp6 7967	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩ ∧ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
6		fvco 6932	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
7		fvco 6932	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
8	6, 7	opeq12d 4837	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
9	8	eqeq2d 2747	. . . . . . . 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 2821	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ (1st ‘(𝐹‘𝑥)) ∈ 𝑌))
11	7	eleq1d 2821	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍))
12	10, 11	anbi12d 632	. . . . . . . 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 632	. . . . . . 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 290	. . . . . 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 715	. . . . 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 32722	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)
17	16	biantrurd 532	. . . . 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 7953	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
19		fofun 6747	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → Fun 1st )
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 1st
21		funco 6532	. . . . . . . . . 10 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
22	20, 21	mpan 690	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (1st ∘ 𝐹))
23	22	adantr 480	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (1st ∘ 𝐹))
24		ssv 3958	. . . . . . . . . . . 12 ⊢ (𝐹 “ dom 𝐹) ⊆ V
25		fof 6746	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st :V⟶V)
26		fdm 6671	. . . . . . . . . . . . 13 ⊢ (1st :V⟶V → dom 1st = V)
27	18, 25, 26	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 1st = V
28	24, 27	sseqtrri 3983	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 1st
29		ssid 3956	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ dom 𝐹
30		funimass3 6999	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
31	29, 30	mpan2 691	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
32	28, 31	mpbii 233	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 1st ))
33	32	sselda 3933	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 1st ))
34		dmco 6213	. . . . . . . . 9 ⊢ dom (1st ∘ 𝐹) = (◡𝐹 “ dom 1st )
35	33, 34	eleqtrrdi 2847	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1st ∘ 𝐹))
36		fvimacnv 6998	. . . . . . . 8 ⊢ ((Fun (1st ∘ 𝐹) ∧ 𝑥 ∈ dom (1st ∘ 𝐹)) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
37	23, 35, 36	syl2anc 584	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
38		fo2nd 7954	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
39		fofun 6747	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → Fun 2nd )
40	38, 39	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 2nd
41		funco 6532	. . . . . . . . . 10 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
42	40, 41	mpan 690	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (2nd ∘ 𝐹))
43	42	adantr 480	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (2nd ∘ 𝐹))
44		fof 6746	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
45		fdm 6671	. . . . . . . . . . . . 13 ⊢ (2nd :V⟶V → dom 2nd = V)
46	38, 44, 45	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 2nd = V
47	24, 46	sseqtrri 3983	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 2nd
48		funimass3 6999	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
49	29, 48	mpan2 691	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
50	47, 49	mpbii 233	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 2nd ))
51	50	sselda 3933	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 2nd ))
52		dmco 6213	. . . . . . . . 9 ⊢ dom (2nd ∘ 𝐹) = (◡𝐹 “ dom 2nd )
53	51, 52	eleqtrrdi 2847	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2nd ∘ 𝐹))
54		fvimacnv 6998	. . . . . . . 8 ⊢ ((Fun (2nd ∘ 𝐹) ∧ 𝑥 ∈ dom (2nd ∘ 𝐹)) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
55	43, 53, 54	syl2anc 584	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
56	37, 55	anbi12d 632	. . . . . 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 715	. . . . 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 307	. . . 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 3405	. . 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 2771	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))})
61		dfin5 3909	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)}
62		dfin5 3909	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)}
63	61, 62	ineq12i 4170	. . 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 6041	. . . . . 6 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom (1st ∘ 𝐹)
65		dmcoss 5924	. . . . . 6 ⊢ dom (1st ∘ 𝐹) ⊆ dom 𝐹
66	64, 65	sstri 3943	. . . . 5 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹
67		sseqin2 4175	. . . . 5 ⊢ ((◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌))
68	66, 67	mpbi 230	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌)
69		cnvimass 6041	. . . . . 6 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom (2nd ∘ 𝐹)
70		dmcoss 5924	. . . . . 6 ⊢ dom (2nd ∘ 𝐹) ⊆ dom 𝐹
71	69, 70	sstri 3943	. . . . 5 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹
72		sseqin2 4175	. . . . 5 ⊢ ((◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍))
73	71, 72	mpbi 230	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍)
74	68, 73	ineq12i 4170	. . 3 ⊢ ((dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍))) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
75		inrab 4268	. . 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 2768	. 2 ⊢ {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))} = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
77	60, 76	eqtrdi 2787	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ⟨cop 4586   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627   ∘ ccom 5628  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7931  2^nd c2nd 7932

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7933  df-2nd 7934

This theorem is referenced by:  xppreima2  32729