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 29835
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 6107 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6549 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6549 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5313 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 578 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6574 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
109frnd 6230 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11 xpss 5293 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1210, 11syl6ss 3773 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
13 xppreima 29834 . . 3 ((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
142, 12, 13sylancr 581 . 2 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)))
15 fo1st 7386 . . . . . . . . 9 1st :V–onto→V
16 fofn 6300 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1715, 16ax-mp 5 . . . . . . . 8 1st Fn V
18 opex 5088 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
1918, 1fnmpti 6200 . . . . . . . 8 𝐻 Fn 𝐴
20 ssv 3785 . . . . . . . 8 ran 𝐻 ⊆ V
21 fnco 6177 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2217, 19, 20, 21mp3an 1585 . . . . . . 7 (1st𝐻) Fn 𝐴
2322a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
243ffnd 6224 . . . . . 6 (𝜑𝐹 Fn 𝐴)
252a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → Fun 𝐻)
2612adantr 472 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran 𝐻 ⊆ (V × V))
27 simpr 477 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝑥𝐴)
2818, 1dmmpti 6201 . . . . . . . . . . 11 dom 𝐻 = 𝐴
2927, 28syl6eleqr 2855 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
30 opfv 29833 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3125, 26, 29, 30syl21anc 866 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
321fvmpt2 6480 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3327, 8, 32syl2anc 579 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3431, 33eqtr3d 2801 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
35 fvex 6388 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
36 fvex 6388 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3735, 36opth 5100 . . . . . . . 8 (⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩ ↔ (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3834, 37sylib 209 . . . . . . 7 ((𝜑𝑥𝐴) → (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3938simpld 488 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4023, 24, 39eqfnfvd 6504 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4140cnveqd 5466 . . . 4 (𝜑(1st𝐻) = 𝐹)
4241imaeq1d 5647 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
43 fo2nd 7387 . . . . . . . . 9 2nd :V–onto→V
44 fofn 6300 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4543, 44ax-mp 5 . . . . . . . 8 2nd Fn V
46 fnco 6177 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4745, 19, 20, 46mp3an 1585 . . . . . . 7 (2nd𝐻) Fn 𝐴
4847a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
495ffnd 6224 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5038simprd 489 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5148, 49, 50eqfnfvd 6504 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5251cnveqd 5466 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5352imaeq1d 5647 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5442, 53ineq12d 3977 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5514, 54eqtrd 2799 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  wss 3732  cop 4340  cmpt 4888   × cxp 5275  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  ccom 5281  Fun wfun 6062   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  1st c1st 7364  2nd c2nd 7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367
This theorem is referenced by:  mbfmco2  30709
  Copyright terms: Public domain W3C validator