Theorem xppreima2 30515

Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)

Hypotheses

Ref	Expression
xppreima2.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
xppreima2.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
xppreima2.3	⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)

Assertion

Ref	Expression
xppreima2	⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝜑,𝑥

Allowed substitution hints:   𝑌(𝑥)   𝑍(𝑥)

Proof of Theorem xppreima2

Step	Hyp	Ref	Expression
1		xppreima2.3	. . . 4 ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
2	1	funmpt2 6378	. . 3 ⊢ Fun 𝐻
3		xppreima2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffvelrnda 6847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
5		xppreima2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
6	5	ffvelrnda 6847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
7		opelxp 5563	. . . . . . 7 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐶))
8	4, 6, 7	sylanbrc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
9	8, 1	fmptd 6874	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶(𝐵 × 𝐶))
10	9	frnd 6509	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11		xpss 5543	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
12	10, 11	sstrdi 3906	. . 3 ⊢ (𝜑 → ran 𝐻 ⊆ (V × V))
13		xppreima 30510	. . 3 ⊢ ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
14	2, 12, 13	sylancr 590	. 2 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
15		fo1st 7718	. . . . . . . . 9 ⊢ 1st :V–onto→V
16		fofn 6582	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
18		opex 5327	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
19	18, 1	fnmpti 6478	. . . . . . . 8 ⊢ 𝐻 Fn 𝐴
20		ssv 3918	. . . . . . . 8 ⊢ ran 𝐻 ⊆ V
21		fnco 6452	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st ∘ 𝐻) Fn 𝐴)
22	17, 19, 20, 21	mp3an 1458	. . . . . . 7 ⊢ (1st ∘ 𝐻) Fn 𝐴
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (1st ∘ 𝐻) Fn 𝐴)
24	3	ffnd 6503	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
25	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun 𝐻)
26	12	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ (V × V))
27		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	18, 1	dmmpti 6479	. . . . . . . . . . 11 ⊢ dom 𝐻 = 𝐴
29	27, 28	eleqtrrdi 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐻)
30		opfv 30509	. . . . . . . . . 10 ⊢ (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
31	25, 26, 29, 30	syl21anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
32	1	fvmpt2 6774	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
33	27, 8, 32	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
34	31, 33	eqtr3d 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
35		fvex 6675	. . . . . . . . 9 ⊢ ((1st ∘ 𝐻)‘𝑥) ∈ V
36		fvex 6675	. . . . . . . . 9 ⊢ ((2nd ∘ 𝐻)‘𝑥) ∈ V
37	35, 36	opth 5339	. . . . . . . 8 ⊢ (⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ↔ (((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥) ∧ ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥)))
38	34, 37	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥) ∧ ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥)))
39	38	simpld 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥))
40	23, 24, 39	eqfnfvd 6800	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝐻) = 𝐹)
41	40	cnveqd 5720	. . . 4 ⊢ (𝜑 → ◡(1st ∘ 𝐻) = ◡𝐹)
42	41	imaeq1d 5904	. . 3 ⊢ (𝜑 → (◡(1st ∘ 𝐻) “ 𝑌) = (◡𝐹 “ 𝑌))
43		fo2nd 7719	. . . . . . . . 9 ⊢ 2nd :V–onto→V
44		fofn 6582	. . . . . . . . 9 ⊢ (2nd :V–onto→V → 2nd Fn V)
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ 2nd Fn V
46		fnco 6452	. . . . . . . 8 ⊢ ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd ∘ 𝐻) Fn 𝐴)
47	45, 19, 20, 46	mp3an 1458	. . . . . . 7 ⊢ (2nd ∘ 𝐻) Fn 𝐴
48	47	a1i 11	. . . . . 6 ⊢ (𝜑 → (2nd ∘ 𝐻) Fn 𝐴)
49	5	ffnd 6503	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
50	38	simprd 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥))
51	48, 49, 50	eqfnfvd 6800	. . . . 5 ⊢ (𝜑 → (2nd ∘ 𝐻) = 𝐺)
52	51	cnveqd 5720	. . . 4 ⊢ (𝜑 → ◡(2nd ∘ 𝐻) = ◡𝐺)
53	52	imaeq1d 5904	. . 3 ⊢ (𝜑 → (◡(2nd ∘ 𝐻) “ 𝑍) = (◡𝐺 “ 𝑍))
54	42, 53	ineq12d 4120	. 2 ⊢ (𝜑 → ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
55	14, 54	eqtrd 2793	1 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  ⟨cop 4531   ↦ cmpt 5115   × cxp 5525  ^◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   ∘ ccom 5531  Fun wfun 6333   Fn wfn 6334  ⟶wf 6335  –onto→wfo 6337  ‘cfv 6339  1^st c1st 7696  2^nd c2nd 7697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-1st 7698  df-2nd 7699

This theorem is referenced by:  mbfmco2  31755