Theorem xppreima2 31863

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 6584	. . 3 ⊢ Fun 𝐻
3		xppreima2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
5		xppreima2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
6	5	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
7		opelxp 5711	. . . . . . 7 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐶))
8	4, 6, 7	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
9	8, 1	fmptd 7110	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶(𝐵 × 𝐶))
10	9	frnd 6722	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11		xpss 5691	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
12	10, 11	sstrdi 3993	. . 3 ⊢ (𝜑 → ran 𝐻 ⊆ (V × V))
13		xppreima 31858	. . 3 ⊢ ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
14	2, 12, 13	sylancr 587	. 2 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
15		fo1st 7991	. . . . . . . . 9 ⊢ 1st :V–onto→V
16		fofn 6804	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
18		opex 5463	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
19	18, 1	fnmpti 6690	. . . . . . . 8 ⊢ 𝐻 Fn 𝐴
20		ssv 4005	. . . . . . . 8 ⊢ ran 𝐻 ⊆ V
21		fnco 6664	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st ∘ 𝐻) Fn 𝐴)
22	17, 19, 20, 21	mp3an 1461	. . . . . . 7 ⊢ (1st ∘ 𝐻) Fn 𝐴
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (1st ∘ 𝐻) Fn 𝐴)
24	3	ffnd 6715	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
25	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun 𝐻)
26	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ (V × V))
27		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	18, 1	dmmpti 6691	. . . . . . . . . . 11 ⊢ dom 𝐻 = 𝐴
29	27, 28	eleqtrrdi 2844	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐻)
30		opfv 31857	. . . . . . . . . 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 7006	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
33	27, 8, 32	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
34	31, 33	eqtr3d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
35		fvex 6901	. . . . . . . . 9 ⊢ ((1st ∘ 𝐻)‘𝑥) ∈ V
36		fvex 6901	. . . . . . . . 9 ⊢ ((2nd ∘ 𝐻)‘𝑥) ∈ V
37	35, 36	opth 5475	. . . . . . . 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 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥) ∧ ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥)))
39	38	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥))
40	23, 24, 39	eqfnfvd 7032	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝐻) = 𝐹)
41	40	cnveqd 5873	. . . 4 ⊢ (𝜑 → ◡(1st ∘ 𝐻) = ◡𝐹)
42	41	imaeq1d 6056	. . 3 ⊢ (𝜑 → (◡(1st ∘ 𝐻) “ 𝑌) = (◡𝐹 “ 𝑌))
43		fo2nd 7992	. . . . . . . . 9 ⊢ 2nd :V–onto→V
44		fofn 6804	. . . . . . . . 9 ⊢ (2nd :V–onto→V → 2nd Fn V)
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ 2nd Fn V
46		fnco 6664	. . . . . . . 8 ⊢ ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd ∘ 𝐻) Fn 𝐴)
47	45, 19, 20, 46	mp3an 1461	. . . . . . 7 ⊢ (2nd ∘ 𝐻) Fn 𝐴
48	47	a1i 11	. . . . . 6 ⊢ (𝜑 → (2nd ∘ 𝐻) Fn 𝐴)
49	5	ffnd 6715	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
50	38	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥))
51	48, 49, 50	eqfnfvd 7032	. . . . 5 ⊢ (𝜑 → (2nd ∘ 𝐻) = 𝐺)
52	51	cnveqd 5873	. . . 4 ⊢ (𝜑 → ◡(2nd ∘ 𝐻) = ◡𝐺)
53	52	imaeq1d 6056	. . 3 ⊢ (𝜑 → (◡(2nd ∘ 𝐻) “ 𝑍) = (◡𝐺 “ 𝑍))
54	42, 53	ineq12d 4212	. 2 ⊢ (𝜑 → ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
55	14, 54	eqtrd 2772	1 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946   ⊆ wss 3947  ⟨cop 4633   ↦ cmpt 5230   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   “ cima 5678   ∘ ccom 5679  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540  1^st c1st 7969  2^nd c2nd 7970

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7971  df-2nd 7972

This theorem is referenced by:  mbfmco2  33252