Theorem xppreima2 29450

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 5927	. . 3 ⊢ Fun 𝐻
3		xppreima2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
5		xppreima2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
6	5	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
7		opelxp 5146	. . . . . . 7 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐶))
8	4, 6, 7	sylanbrc 698	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
9	8, 1	fmptd 6385	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶(𝐵 × 𝐶))
10		frn 6053	. . . . 5 ⊢ (𝐻:𝐴⟶(𝐵 × 𝐶) → ran 𝐻 ⊆ (𝐵 × 𝐶))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
12		xpss 5226	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
13	11, 12	syl6ss 3615	. . 3 ⊢ (𝜑 → ran 𝐻 ⊆ (V × V))
14		xppreima 29449	. . 3 ⊢ ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
15	2, 13, 14	sylancr 695	. 2 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
16		fo1st 7188	. . . . . . . . 9 ⊢ 1st :V–onto→V
17		fofn 6117	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
19		opex 4932	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
20	19, 1	fnmpti 6022	. . . . . . . 8 ⊢ 𝐻 Fn 𝐴
21		ssv 3625	. . . . . . . 8 ⊢ ran 𝐻 ⊆ V
22		fnco 5999	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st ∘ 𝐻) Fn 𝐴)
23	18, 20, 21, 22	mp3an 1424	. . . . . . 7 ⊢ (1st ∘ 𝐻) Fn 𝐴
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (1st ∘ 𝐻) Fn 𝐴)
25		ffn 6045	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
26	3, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun 𝐻)
28	13	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ (V × V))
29		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
30	19, 1	dmmpti 6023	. . . . . . . . . . 11 ⊢ dom 𝐻 = 𝐴
31	29, 30	syl6eleqr 2712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐻)
32		opfv 29448	. . . . . . . . . 10 ⊢ (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
33	27, 28, 31, 32	syl21anc 1325	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
34	1	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
35	29, 8, 34	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
36	33, 35	eqtr3d 2658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
37		fvex 6201	. . . . . . . . 9 ⊢ ((1st ∘ 𝐻)‘𝑥) ∈ V
38		fvex 6201	. . . . . . . . 9 ⊢ ((2nd ∘ 𝐻)‘𝑥) ∈ V
39	37, 38	opth 4945	. . . . . . . 8 ⊢ (⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ↔ (((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥) ∧ ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥)))
40	36, 39	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥) ∧ ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥)))
41	40	simpld 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥))
42	24, 26, 41	eqfnfvd 6314	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝐻) = 𝐹)
43	42	cnveqd 5298	. . . 4 ⊢ (𝜑 → ◡(1st ∘ 𝐻) = ◡𝐹)
44	43	imaeq1d 5465	. . 3 ⊢ (𝜑 → (◡(1st ∘ 𝐻) “ 𝑌) = (◡𝐹 “ 𝑌))
45		fo2nd 7189	. . . . . . . . 9 ⊢ 2nd :V–onto→V
46		fofn 6117	. . . . . . . . 9 ⊢ (2nd :V–onto→V → 2nd Fn V)
47	45, 46	ax-mp 5	. . . . . . . 8 ⊢ 2nd Fn V
48		fnco 5999	. . . . . . . 8 ⊢ ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd ∘ 𝐻) Fn 𝐴)
49	47, 20, 21, 48	mp3an 1424	. . . . . . 7 ⊢ (2nd ∘ 𝐻) Fn 𝐴
50	49	a1i 11	. . . . . 6 ⊢ (𝜑 → (2nd ∘ 𝐻) Fn 𝐴)
51		ffn 6045	. . . . . . 7 ⊢ (𝐺:𝐴⟶𝐶 → 𝐺 Fn 𝐴)
52	5, 51	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
53	40	simprd 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥))
54	50, 52, 53	eqfnfvd 6314	. . . . 5 ⊢ (𝜑 → (2nd ∘ 𝐻) = 𝐺)
55	54	cnveqd 5298	. . . 4 ⊢ (𝜑 → ◡(2nd ∘ 𝐻) = ◡𝐺)
56	55	imaeq1d 5465	. . 3 ⊢ (𝜑 → (◡(2nd ∘ 𝐻) “ 𝑍) = (◡𝐺 “ 𝑍))
57	44, 56	ineq12d 3815	. 2 ⊢ (𝜑 → ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
58	15, 57	eqtrd 2656	1 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ⟨cop 4183   ↦ cmpt 4729   × 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-csb 3534  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:  mbfmco2  30327