Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xppreima2 Structured version   Visualization version   GIF version

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  –onto→wfo 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
 Copyright terms: Public domain W3C validator