Theorem xppreima2 30413

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 6363	. . 3 ⊢ Fun 𝐻
3		xppreima2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
5		xppreima2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
6	5	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
7		opelxp 5555	. . . . . . 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 6855	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶(𝐵 × 𝐶))
10	9	frnd 6494	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11		xpss 5535	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
12	10, 11	sstrdi 3927	. . 3 ⊢ (𝜑 → ran 𝐻 ⊆ (V × V))
13		xppreima 30408	. . 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 7691	. . . . . . . . 9 ⊢ 1st :V–onto→V
16		fofn 6567	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
18		opex 5321	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
19	18, 1	fnmpti 6463	. . . . . . . 8 ⊢ 𝐻 Fn 𝐴
20		ssv 3939	. . . . . . . 8 ⊢ ran 𝐻 ⊆ V
21		fnco 6437	. . . . . . . 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 6488	. . . . . 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 6464	. . . . . . . . . . 11 ⊢ dom 𝐻 = 𝐴
29	27, 28	eleqtrrdi 2901	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐻)
30		opfv 30407	. . . . . . . . . 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 6756	. . . . . . . . . 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 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
35		fvex 6658	. . . . . . . . 9 ⊢ ((1st ∘ 𝐻)‘𝑥) ∈ V
36		fvex 6658	. . . . . . . . 9 ⊢ ((2nd ∘ 𝐻)‘𝑥) ∈ V
37	35, 36	opth 5333	. . . . . . . 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 6782	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝐻) = 𝐹)
41	40	cnveqd 5710	. . . 4 ⊢ (𝜑 → ◡(1st ∘ 𝐻) = ◡𝐹)
42	41	imaeq1d 5895	. . 3 ⊢ (𝜑 → (◡(1st ∘ 𝐻) “ 𝑌) = (◡𝐹 “ 𝑌))
43		fo2nd 7692	. . . . . . . . 9 ⊢ 2nd :V–onto→V
44		fofn 6567	. . . . . . . . 9 ⊢ (2nd :V–onto→V → 2nd Fn V)
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ 2nd Fn V
46		fnco 6437	. . . . . . . 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 6488	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
50	38	simprd 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥))
51	48, 49, 50	eqfnfvd 6782	. . . . 5 ⊢ (𝜑 → (2nd ∘ 𝐻) = 𝐺)
52	51	cnveqd 5710	. . . 4 ⊢ (𝜑 → ◡(2nd ∘ 𝐻) = ◡𝐺)
53	52	imaeq1d 5895	. . 3 ⊢ (𝜑 → (◡(2nd ∘ 𝐻) “ 𝑍) = (◡𝐺 “ 𝑍))
54	42, 53	ineq12d 4140	. 2 ⊢ (𝜑 → ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
55	14, 54	eqtrd 2833	1 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ⟨cop 4531   ↦ cmpt 5110   × cxp 5517  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522   ∘ ccom 5523  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –onto→wfo 6322  ‘cfv 6324  1^st c1st 7669  2^nd c2nd 7670

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 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672

This theorem is referenced by:  mbfmco2  31633