Theorem xppreima2 32345

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 6577	. . 3 ⊢ Fun 𝐻
3		xppreima2.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffvelcdmda 7076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
5		xppreima2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
6	5	ffvelcdmda 7076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐶)
7		opelxp 5702	. . . . . . 7 ⊢ (⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐶))
8	4, 6, 7	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶))
9	8, 1	fmptd 7105	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶(𝐵 × 𝐶))
10	9	frnd 6715	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11		xpss 5682	. . . 4 ⊢ (𝐵 × 𝐶) ⊆ (V × V)
12	10, 11	sstrdi 3986	. . 3 ⊢ (𝜑 → ran 𝐻 ⊆ (V × V))
13		xppreima 32340	. . 3 ⊢ ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
14	2, 12, 13	sylancr 586	. 2 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)))
15		fo1st 7988	. . . . . . . . 9 ⊢ 1st :V–onto→V
16		fofn 6797	. . . . . . . . 9 ⊢ (1st :V–onto→V → 1st Fn V)
17	15, 16	ax-mp 5	. . . . . . . 8 ⊢ 1st Fn V
18		opex 5454	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ V
19	18, 1	fnmpti 6683	. . . . . . . 8 ⊢ 𝐻 Fn 𝐴
20		ssv 3998	. . . . . . . 8 ⊢ ran 𝐻 ⊆ V
21		fnco 6657	. . . . . . . 8 ⊢ ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st ∘ 𝐻) Fn 𝐴)
22	17, 19, 20, 21	mp3an 1457	. . . . . . 7 ⊢ (1st ∘ 𝐻) Fn 𝐴
23	22	a1i 11	. . . . . 6 ⊢ (𝜑 → (1st ∘ 𝐻) Fn 𝐴)
24	3	ffnd 6708	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
25	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun 𝐻)
26	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝐻 ⊆ (V × V))
27		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
28	18, 1	dmmpti 6684	. . . . . . . . . . 11 ⊢ dom 𝐻 = 𝐴
29	27, 28	eleqtrrdi 2836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐻)
30		opfv 32339	. . . . . . . . . 10 ⊢ (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
31	25, 26, 29, 30	syl21anc 835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩)
32	1	fvmpt2 6999	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
33	27, 8, 32	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
34	31, 33	eqtr3d 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨((1st ∘ 𝐻)‘𝑥), ((2nd ∘ 𝐻)‘𝑥)⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
35		fvex 6894	. . . . . . . . 9 ⊢ ((1st ∘ 𝐻)‘𝑥) ∈ V
36		fvex 6894	. . . . . . . . 9 ⊢ ((2nd ∘ 𝐻)‘𝑥) ∈ V
37	35, 36	opth 5466	. . . . . . . 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 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((1st ∘ 𝐻)‘𝑥) = (𝐹‘𝑥))
40	23, 24, 39	eqfnfvd 7025	. . . . 5 ⊢ (𝜑 → (1st ∘ 𝐻) = 𝐹)
41	40	cnveqd 5865	. . . 4 ⊢ (𝜑 → ◡(1st ∘ 𝐻) = ◡𝐹)
42	41	imaeq1d 6048	. . 3 ⊢ (𝜑 → (◡(1st ∘ 𝐻) “ 𝑌) = (◡𝐹 “ 𝑌))
43		fo2nd 7989	. . . . . . . . 9 ⊢ 2nd :V–onto→V
44		fofn 6797	. . . . . . . . 9 ⊢ (2nd :V–onto→V → 2nd Fn V)
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ 2nd Fn V
46		fnco 6657	. . . . . . . 8 ⊢ ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd ∘ 𝐻) Fn 𝐴)
47	45, 19, 20, 46	mp3an 1457	. . . . . . 7 ⊢ (2nd ∘ 𝐻) Fn 𝐴
48	47	a1i 11	. . . . . 6 ⊢ (𝜑 → (2nd ∘ 𝐻) Fn 𝐴)
49	5	ffnd 6708	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐴)
50	38	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((2nd ∘ 𝐻)‘𝑥) = (𝐺‘𝑥))
51	48, 49, 50	eqfnfvd 7025	. . . . 5 ⊢ (𝜑 → (2nd ∘ 𝐻) = 𝐺)
52	51	cnveqd 5865	. . . 4 ⊢ (𝜑 → ◡(2nd ∘ 𝐻) = ◡𝐺)
53	52	imaeq1d 6048	. . 3 ⊢ (𝜑 → (◡(2nd ∘ 𝐻) “ 𝑍) = (◡𝐺 “ 𝑍))
54	42, 53	ineq12d 4205	. 2 ⊢ (𝜑 → ((◡(1st ∘ 𝐻) “ 𝑌) ∩ (◡(2nd ∘ 𝐻) “ 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
55	14, 54	eqtrd 2764	1 ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ⟨cop 4626   ↦ cmpt 5221   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   “ cima 5669   ∘ ccom 5670  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  1^st c1st 7966  2^nd c2nd 7967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-1st 7968  df-2nd 7969

This theorem is referenced by:  mbfmco2  33753