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Theorem xppreima2 28830
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 5841 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5070 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 695 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6292 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
10 frn 5966 . . . . 5 (𝐻:𝐴⟶(𝐵 × 𝐶) → ran 𝐻 ⊆ (𝐵 × 𝐶))
119, 10syl 17 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
12 xpss 5149 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1311, 12syl6ss 3580 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
14 xppreima 28829 . . 3 ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
152, 13, 14sylancr 694 . 2 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
16 fo1st 7079 . . . . . . . . 9 1st :V–onto→V
17 fofn 6030 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1816, 17ax-mp 5 . . . . . . . 8 1st Fn V
19 opex 4859 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
2019, 1fnmpti 5935 . . . . . . . 8 𝐻 Fn 𝐴
21 ssv 3588 . . . . . . . 8 ran 𝐻 ⊆ V
22 fnco 5913 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2318, 20, 21, 22mp3an 1416 . . . . . . 7 (1st𝐻) Fn 𝐴
2423a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
25 ffn 5958 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
263, 25syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
272a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
2813adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ (V × V))
29 simpr 476 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3019, 1dmmpti 5936 . . . . . . . . . . 11 dom 𝐻 = 𝐴
3129, 30syl6eleqr 2699 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
32 opfv 28828 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3327, 28, 31, 32syl21anc 1317 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
341fvmpt2 6200 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3529, 8, 34syl2anc 691 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3633, 35eqtr3d 2646 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
37 fvex 6113 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
38 fvex 6113 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3937, 38opth 4871 . . . . . . . 8 (⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩ ↔ (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
4036, 39sylib 207 . . . . . . 7 ((𝜑𝑥𝐴) → (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
4140simpld 474 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4224, 26, 41eqfnfvd 6222 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4342cnveqd 5220 . . . 4 (𝜑(1st𝐻) = 𝐹)
4443imaeq1d 5384 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
45 fo2nd 7080 . . . . . . . . 9 2nd :V–onto→V
46 fofn 6030 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4745, 46ax-mp 5 . . . . . . . 8 2nd Fn V
48 fnco 5913 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4947, 20, 21, 48mp3an 1416 . . . . . . 7 (2nd𝐻) Fn 𝐴
5049a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
51 ffn 5958 . . . . . . 7 (𝐺:𝐴𝐶𝐺 Fn 𝐴)
525, 51syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5340simprd 478 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5450, 52, 53eqfnfvd 6222 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5554cnveqd 5220 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5655imaeq1d 5384 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5744, 56ineq12d 3777 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5815, 57eqtrd 2644 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  cop 4131  cmpt 4643   × cxp 5036  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  ccom 5042  Fun wfun 5798   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  1st c1st 7057  2nd c2nd 7058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060
This theorem is referenced by:  mbfmco2  29654
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