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Theorem xppreima2 29289
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 5885 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6315 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6315 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5106 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 697 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6340 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
10 frn 6010 . . . . 5 (𝐻:𝐴⟶(𝐵 × 𝐶) → ran 𝐻 ⊆ (𝐵 × 𝐶))
119, 10syl 17 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
12 xpss 5187 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1311, 12syl6ss 3595 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
14 xppreima 29288 . . 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 7133 . . . . . . . . 9 1st :V–onto→V
17 fofn 6074 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1816, 17ax-mp 5 . . . . . . . 8 1st Fn V
19 opex 4893 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
2019, 1fnmpti 5979 . . . . . . . 8 𝐻 Fn 𝐴
21 ssv 3604 . . . . . . . 8 ran 𝐻 ⊆ V
22 fnco 5957 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2318, 20, 21, 22mp3an 1421 . . . . . . 7 (1st𝐻) Fn 𝐴
2423a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
25 ffn 6002 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
263, 25syl 17 . . . . . 6 (𝜑𝐹 Fn 𝐴)
272a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
2813adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ (V × V))
29 simpr 477 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
3019, 1dmmpti 5980 . . . . . . . . . . 11 dom 𝐻 = 𝐴
3129, 30syl6eleqr 2709 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
32 opfv 29287 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3327, 28, 31, 32syl21anc 1322 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
341fvmpt2 6248 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3529, 8, 34syl2anc 692 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3633, 35eqtr3d 2657 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
37 fvex 6158 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
38 fvex 6158 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3937, 38opth 4905 . . . . . . . 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 475 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4224, 26, 41eqfnfvd 6270 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4342cnveqd 5258 . . . 4 (𝜑(1st𝐻) = 𝐹)
4443imaeq1d 5424 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
45 fo2nd 7134 . . . . . . . . 9 2nd :V–onto→V
46 fofn 6074 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4745, 46ax-mp 5 . . . . . . . 8 2nd Fn V
48 fnco 5957 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4947, 20, 21, 48mp3an 1421 . . . . . . 7 (2nd𝐻) Fn 𝐴
5049a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
51 ffn 6002 . . . . . . 7 (𝐺:𝐴𝐶𝐺 Fn 𝐴)
525, 51syl 17 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5340simprd 479 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5450, 52, 53eqfnfvd 6270 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5554cnveqd 5258 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5655imaeq1d 5424 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5744, 56ineq12d 3793 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5815, 57eqtrd 2655 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  cop 4154  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  Fun wfun 5841   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  mbfmco2  30105
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