Theorem xppreima 29449

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 5918	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		fncnvima2 6339	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
3	1, 2	sylbi 207	. . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
4	3	adantr 481	. . 3 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) ∈ (𝑌 × 𝑍)})
5		fvco 6274	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥)))
6		fvco 6274	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥)))
7	5, 6	opeq12d 4410	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
8	7	eqeq2d 2632	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩ ↔ (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
9	5	eleq1d 2686	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ (1st ‘(𝐹‘𝑥)) ∈ 𝑌))
10	6	eleq1d 2686	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍))
11	9, 10	anbi12d 747	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍) ↔ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
12	8, 11	anbi12d 747	. . . . . . 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))))
13		elxp6 7200	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩ ∧ ((1st ‘(𝐹‘𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹‘𝑥)) ∈ 𝑍)))
14	12, 13	syl6rbbr 279	. . . . . 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 751	. . . . 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 29448	. . . . . 6 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)
17	16	biantrurd 529	. . . . 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 7188	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
19		fofun 6116	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → Fun 1st )
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 1st
21		funco 5928	. . . . . . . . . 10 ⊢ ((Fun 1st ∧ Fun 𝐹) → Fun (1st ∘ 𝐹))
22	20, 21	mpan 706	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (1st ∘ 𝐹))
23	22	adantr 481	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (1st ∘ 𝐹))
24		ssv 3625	. . . . . . . . . . . 12 ⊢ (𝐹 “ dom 𝐹) ⊆ V
25		fof 6115	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → 1st :V⟶V)
26		fdm 6051	. . . . . . . . . . . . 13 ⊢ (1st :V⟶V → dom 1st = V)
27	18, 25, 26	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 1st = V
28	24, 27	sseqtr4i 3638	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 1st
29		ssid 3624	. . . . . . . . . . . 12 ⊢ dom 𝐹 ⊆ dom 𝐹
30		funimass3 6333	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
31	29, 30	mpan2 707	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 1st )))
32	28, 31	mpbii 223	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 1st ))
33	32	sselda 3603	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 1st ))
34		dmco 5643	. . . . . . . . 9 ⊢ dom (1st ∘ 𝐹) = (◡𝐹 “ dom 1st )
35	33, 34	syl6eleqr 2712	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1st ∘ 𝐹))
36		fvimacnv 6332	. . . . . . . 8 ⊢ ((Fun (1st ∘ 𝐹) ∧ 𝑥 ∈ dom (1st ∘ 𝐹)) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
37	23, 35, 36	syl2anc 693	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((1st ∘ 𝐹)‘𝑥) ∈ 𝑌 ↔ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)))
38		fo2nd 7189	. . . . . . . . . . 11 ⊢ 2nd :V–onto→V
39		fofun 6116	. . . . . . . . . . 11 ⊢ (2nd :V–onto→V → Fun 2nd )
40	38, 39	ax-mp 5	. . . . . . . . . 10 ⊢ Fun 2nd
41		funco 5928	. . . . . . . . . 10 ⊢ ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd ∘ 𝐹))
42	40, 41	mpan 706	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (2nd ∘ 𝐹))
43	42	adantr 481	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → Fun (2nd ∘ 𝐹))
44		fof 6115	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
45		fdm 6051	. . . . . . . . . . . . 13 ⊢ (2nd :V⟶V → dom 2nd = V)
46	38, 44, 45	mp2b 10	. . . . . . . . . . . 12 ⊢ dom 2nd = V
47	24, 46	sseqtr4i 3638	. . . . . . . . . . 11 ⊢ (𝐹 “ dom 𝐹) ⊆ dom 2nd
48		funimass3 6333	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
49	29, 48	mpan2 707	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (◡𝐹 “ dom 2nd )))
50	47, 49	mpbii 223	. . . . . . . . . 10 ⊢ (Fun 𝐹 → dom 𝐹 ⊆ (◡𝐹 “ dom 2nd ))
51	50	sselda 3603	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ (◡𝐹 “ dom 2nd ))
52		dmco 5643	. . . . . . . . 9 ⊢ dom (2nd ∘ 𝐹) = (◡𝐹 “ dom 2nd )
53	51, 52	syl6eleqr 2712	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2nd ∘ 𝐹))
54		fvimacnv 6332	. . . . . . . 8 ⊢ ((Fun (2nd ∘ 𝐹) ∧ 𝑥 ∈ dom (2nd ∘ 𝐹)) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
55	43, 53, 54	syl2anc 693	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((2nd ∘ 𝐹)‘𝑥) ∈ 𝑍 ↔ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)))
56	37, 55	anbi12d 747	. . . . . 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 751	. . . . 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 296	. . . 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 3188	. . 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 2656	. 2 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))})
61		dfin5 3582	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌)}
62		dfin5 3582	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍)}
63	61, 62	ineq12i 3812	. . 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 5485	. . . . . 6 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom (1st ∘ 𝐹)
65		dmcoss 5385	. . . . . 6 ⊢ dom (1st ∘ 𝐹) ⊆ dom 𝐹
66	64, 65	sstri 3612	. . . . 5 ⊢ (◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹
67		sseqin2 3817	. . . . 5 ⊢ ((◡(1st ∘ 𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌))
68	66, 67	mpbi 220	. . . 4 ⊢ (dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) = (◡(1st ∘ 𝐹) “ 𝑌)
69		cnvimass 5485	. . . . . 6 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom (2nd ∘ 𝐹)
70		dmcoss 5385	. . . . . 6 ⊢ dom (2nd ∘ 𝐹) ⊆ dom 𝐹
71	69, 70	sstri 3612	. . . . 5 ⊢ (◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹
72		sseqin2 3817	. . . . 5 ⊢ ((◡(2nd ∘ 𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍))
73	71, 72	mpbi 220	. . . 4 ⊢ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍)) = (◡(2nd ∘ 𝐹) “ 𝑍)
74	68, 73	ineq12i 3812	. . 3 ⊢ ((dom 𝐹 ∩ (◡(1st ∘ 𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ (◡(2nd ∘ 𝐹) “ 𝑍))) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
75		inrab 3899	. . 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 2653	. 2 ⊢ {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ (◡(1st ∘ 𝐹) “ 𝑌) ∧ 𝑥 ∈ (◡(2nd ∘ 𝐹) “ 𝑍))} = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍))
77	60, 76	syl6eq 2672	1 ⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (◡𝐹 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ⟨cop 4183   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  xppreima2  29450