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Theorem xppreima2 29759
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 (𝜑𝐹:𝐴𝐵)
xppreima2.2 (𝜑𝐺:𝐴𝐶)
xppreima2.3 𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
Assertion
Ref Expression
xppreima2 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
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
StepHypRef Expression
1 xppreima2.3 . . . 4 𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
21funmpt2 6088 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6522 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6522 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5303 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 701 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6548 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
10 frn 6214 . . . . 5 (𝐻:𝐴⟶(𝐵 × 𝐶) → ran 𝐻 ⊆ (𝐵 × 𝐶))
119, 10syl 17 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
12 xpss 5282 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1311, 12syl6ss 3756 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
14 xppreima 29758 . . 3 ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
152, 13, 14sylancr 698 . 2 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
16 fo1st 7353 . . . . . . . . 9 1st :V–onto→V
17 fofn 6278 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1816, 17ax-mp 5 . . . . . . . 8 1st Fn V
19 opex 5081 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
2019, 1fnmpti 6183 . . . . . . . 8 𝐻 Fn 𝐴
21 ssv 3766 . . . . . . . 8 ran 𝐻 ⊆ V
22 fnco 6160 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2318, 20, 21, 22mp3an 1573 . . . . . . 7 (1st𝐻) Fn 𝐴
2423a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
25 ffn 6206 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
263, 25syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
272a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
2813adantr 472 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ (V × V))
29 simpr 479 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3019, 1dmmpti 6184 . . . . . . . . . . 11 dom 𝐻 = 𝐴
3129, 30syl6eleqr 2850 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
32 opfv 29757 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3327, 28, 31, 32syl21anc 1476 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
341fvmpt2 6453 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3529, 8, 34syl2anc 696 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3633, 35eqtr3d 2796 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
37 fvex 6362 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
38 fvex 6362 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3937, 38opth 5093 . . . . . . . 8 (⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩ ↔ (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
4036, 39sylib 208 . . . . . . 7 ((𝜑𝑥𝐴) → (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
4140simpld 477 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4224, 26, 41eqfnfvd 6477 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4342cnveqd 5453 . . . 4 (𝜑(1st𝐻) = 𝐹)
4443imaeq1d 5623 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
45 fo2nd 7354 . . . . . . . . 9 2nd :V–onto→V
46 fofn 6278 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4745, 46ax-mp 5 . . . . . . . 8 2nd Fn V
48 fnco 6160 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4947, 20, 21, 48mp3an 1573 . . . . . . 7 (2nd𝐻) Fn 𝐴
5049a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
51 ffn 6206 . . . . . . 7 (𝐺:𝐴𝐶𝐺 Fn 𝐴)
525, 51syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5340simprd 482 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5450, 52, 53eqfnfvd 6477 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5554cnveqd 5453 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5655imaeq1d 5623 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5744, 56ineq12d 3958 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5815, 57eqtrd 2794 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  wss 3715  cop 4327  cmpt 4881   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  ccom 5270  Fun wfun 6043   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  1st c1st 7331  2nd c2nd 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334
This theorem is referenced by:  mbfmco2  30636
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