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

Theorem xppreima2 30323
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 6387 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5584 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 583 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6870 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
109frnd 6514 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11 xpss 5564 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1210, 11sstrdi 3976 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
13 xppreima 30322 . . 3 ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
142, 12, 13sylancr 587 . 2 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
15 fo1st 7698 . . . . . . . . 9 1st :V–onto→V
16 fofn 6585 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1715, 16ax-mp 5 . . . . . . . 8 1st Fn V
18 opex 5347 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
1918, 1fnmpti 6484 . . . . . . . 8 𝐻 Fn 𝐴
20 ssv 3988 . . . . . . . 8 ran 𝐻 ⊆ V
21 fnco 6458 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2217, 19, 20, 21mp3an 1452 . . . . . . 7 (1st𝐻) Fn 𝐴
2322a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
243ffnd 6508 . . . . . 6 (𝜑𝐹 Fn 𝐴)
252a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
2612adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ (V × V))
27 simpr 485 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
2818, 1dmmpti 6485 . . . . . . . . . . 11 dom 𝐻 = 𝐴
2927, 28eleqtrrdi 2921 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
30 opfv 30321 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3125, 26, 29, 30syl21anc 833 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
321fvmpt2 6771 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3327, 8, 32syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3431, 33eqtr3d 2855 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
35 fvex 6676 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
36 fvex 6676 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3735, 36opth 5359 . . . . . . . 8 (⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩ ↔ (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3834, 37sylib 219 . . . . . . 7 ((𝜑𝑥𝐴) → (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3938simpld 495 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4023, 24, 39eqfnfvd 6797 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4140cnveqd 5739 . . . 4 (𝜑(1st𝐻) = 𝐹)
4241imaeq1d 5921 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
43 fo2nd 7699 . . . . . . . . 9 2nd :V–onto→V
44 fofn 6585 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4543, 44ax-mp 5 . . . . . . . 8 2nd Fn V
46 fnco 6458 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4745, 19, 20, 46mp3an 1452 . . . . . . 7 (2nd𝐻) Fn 𝐴
4847a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
495ffnd 6508 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5038simprd 496 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5148, 49, 50eqfnfvd 6797 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5251cnveqd 5739 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5352imaeq1d 5921 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5442, 53ineq12d 4187 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5514, 54eqtrd 2853 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  wss 3933  cop 4563  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  mbfmco2  31422
  Copyright terms: Public domain W3C validator