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Theorem xppreima2 30310
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 6391 . . 3 Fun 𝐻
3 xppreima2.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
43ffvelrnda 6847 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
5 xppreima2.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐶)
65ffvelrnda 6847 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐶)
7 opelxp 5590 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐶))
84, 6, 7sylanbrc 583 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶))
98, 1fmptd 6874 . . . . 5 (𝜑𝐻:𝐴⟶(𝐵 × 𝐶))
109frnd 6518 . . . 4 (𝜑 → ran 𝐻 ⊆ (𝐵 × 𝐶))
11 xpss 5570 . . . 4 (𝐵 × 𝐶) ⊆ (V × V)
1210, 11syl6ss 3983 . . 3 (𝜑 → ran 𝐻 ⊆ (V × V))
13 xppreima 30309 . . 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 7700 . . . . . . . . 9 1st :V–onto→V
16 fofn 6589 . . . . . . . . 9 (1st :V–onto→V → 1st Fn V)
1715, 16ax-mp 5 . . . . . . . 8 1st Fn V
18 opex 5353 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
1918, 1fnmpti 6488 . . . . . . . 8 𝐻 Fn 𝐴
20 ssv 3995 . . . . . . . 8 ran 𝐻 ⊆ V
21 fnco 6462 . . . . . . . 8 ((1st Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (1st𝐻) Fn 𝐴)
2217, 19, 20, 21mp3an 1454 . . . . . . 7 (1st𝐻) Fn 𝐴
2322a1i 11 . . . . . 6 (𝜑 → (1st𝐻) Fn 𝐴)
243ffnd 6512 . . . . . 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 6489 . . . . . . . . . . 11 dom 𝐻 = 𝐴
2927, 28syl6eleqr 2929 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ dom 𝐻)
30 opfv 30308 . . . . . . . . . 10 (((Fun 𝐻 ∧ ran 𝐻 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐻) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
3125, 26, 29, 30syl21anc 835 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩)
321fvmpt2 6775 . . . . . . . . . 10 ((𝑥𝐴 ∧ ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐶)) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3327, 8, 32syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐻𝑥) = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
3431, 33eqtr3d 2863 . . . . . . . 8 ((𝜑𝑥𝐴) → ⟨((1st𝐻)‘𝑥), ((2nd𝐻)‘𝑥)⟩ = ⟨(𝐹𝑥), (𝐺𝑥)⟩)
35 fvex 6680 . . . . . . . . 9 ((1st𝐻)‘𝑥) ∈ V
36 fvex 6680 . . . . . . . . 9 ((2nd𝐻)‘𝑥) ∈ V
3735, 36opth 5365 . . . . . . . 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 6801 . . . . 5 (𝜑 → (1st𝐻) = 𝐹)
4140cnveqd 5745 . . . 4 (𝜑(1st𝐻) = 𝐹)
4241imaeq1d 5926 . . 3 (𝜑 → ((1st𝐻) “ 𝑌) = (𝐹𝑌))
43 fo2nd 7701 . . . . . . . . 9 2nd :V–onto→V
44 fofn 6589 . . . . . . . . 9 (2nd :V–onto→V → 2nd Fn V)
4543, 44ax-mp 5 . . . . . . . 8 2nd Fn V
46 fnco 6462 . . . . . . . 8 ((2nd Fn V ∧ 𝐻 Fn 𝐴 ∧ ran 𝐻 ⊆ V) → (2nd𝐻) Fn 𝐴)
4745, 19, 20, 46mp3an 1454 . . . . . . 7 (2nd𝐻) Fn 𝐴
4847a1i 11 . . . . . 6 (𝜑 → (2nd𝐻) Fn 𝐴)
495ffnd 6512 . . . . . 6 (𝜑𝐺 Fn 𝐴)
5038simprd 496 . . . . . 6 ((𝜑𝑥𝐴) → ((2nd𝐻)‘𝑥) = (𝐺𝑥))
5148, 49, 50eqfnfvd 6801 . . . . 5 (𝜑 → (2nd𝐻) = 𝐺)
5251cnveqd 5745 . . . 4 (𝜑(2nd𝐻) = 𝐺)
5352imaeq1d 5926 . . 3 (𝜑 → ((2nd𝐻) “ 𝑍) = (𝐺𝑍))
5442, 53ineq12d 4194 . 2 (𝜑 → (((1st𝐻) “ 𝑌) ∩ ((2nd𝐻) “ 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
5514, 54eqtrd 2861 1 (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cin 3939  wss 3940  cop 4570  cmpt 5143   × cxp 5552  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557  ccom 5558  Fun wfun 6346   Fn wfn 6347  wf 6348  ontowfo 6350  cfv 6352  1st c1st 7678  2nd c2nd 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-1st 7680  df-2nd 7681
This theorem is referenced by:  mbfmco2  31409
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