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Theorem xppreima2 31567
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 6540 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5669 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 583 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 7062 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
109frnd 6676 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11 xpss 5649 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1210, 11sstrdi 3956 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
13 xppreima 31562 . . 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 7941 . . . . . . . . 9 1st :V–onto→V
16 fofn 6758 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1715, 16ax-mp 5 . . . . . . . 8 1st Fn V
18 opex 5421 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
1918, 1fnmpti 6644 . . . . . . . 8 𝐻 Fn 𝐴
20 ssv 3968 . . . . . . . 8 ran 𝐻 ⊆ V
21 fnco 6618 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2217, 19, 20, 21mp3an 1461 . . . . . . 7 (1st𝐻) Fn 𝐴
2322a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
243ffnd 6669 . . . . . 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 6645 . . . . . . . . . . 11 dom 𝐻 = 𝐴
2927, 28eleqtrrdi 2849 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
30 opfv 31561 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3125, 26, 29, 30syl21anc 836 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
321fvmpt2 6959 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3327, 8, 32syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3431, 33eqtr3d 2778 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
35 fvex 6855 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
36 fvex 6855 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3735, 36opth 5433 . . . . . . . 8 (⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩ ↔ (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3834, 37sylib 217 . . . . . . 7 ((𝜑𝑥𝐴) → (((1st𝐻)‘𝑥) = (𝐹𝑥) ∧ ((2nd𝐻)‘𝑥) = (𝐺𝑥)))
3938simpld 495 . . . . . 6 ((𝜑𝑥𝐴) → ((1st𝐻)‘𝑥) = (𝐹𝑥))
4023, 24, 39eqfnfvd 6985 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4140cnveqd 5831 . . . 4 (𝜑(1st𝐻) = 𝐹)
4241imaeq1d 6012 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
43 fo2nd 7942 . . . . . . . . 9 2nd :V–onto→V
44 fofn 6758 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4543, 44ax-mp 5 . . . . . . . 8 2nd Fn V
46 fnco 6618 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4745, 19, 20, 46mp3an 1461 . . . . . . 7 (2nd𝐻) Fn 𝐴
4847a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
495ffnd 6669 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5038simprd 496 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5148, 49, 50eqfnfvd 6985 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5251cnveqd 5831 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5352imaeq1d 6012 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5442, 53ineq12d 4173 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5514, 54eqtrd 2776 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cin 3909  wss 3910  cop 4592  cmpt 5188   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  1st c1st 7919  2nd c2nd 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-1st 7921  df-2nd 7922
This theorem is referenced by:  mbfmco2  32865
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