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Theorem funcf2lem 48930
Description: A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
Assertion
Ref Expression
funcf2lem (𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦))))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧

Proof of Theorem funcf2lem
StepHypRef Expression
1 elixp2 8942 . 2 (𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))))
2 fveq2 6905 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
3 df-ov 7435 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2794 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
5 vex 3483 . . . . . . . . . 10 𝑥 ∈ V
6 vex 3483 . . . . . . . . . 10 𝑦 ∈ V
75, 6op1std 8025 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
87fveq2d 6909 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
95, 6op2ndd 8026 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
109fveq2d 6909 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
118, 10oveq12d 7450 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹𝑥)𝐽(𝐹𝑦)))
12 fveq2 6905 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
13 df-ov 7435 . . . . . . . 8 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
1412, 13eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
1511, 14oveq12d 7450 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
164, 15eleq12d 2834 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦))))
17 ovex 7465 . . . . . 6 ((𝐹𝑥)𝐽(𝐹𝑦)) ∈ V
18 ovex 7465 . . . . . 6 (𝑥𝐻𝑦) ∈ V
1917, 18elmap 8912 . . . . 5 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
2016, 19bitrdi 287 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦))))
2120ralxp 5851 . . 3 (∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
22213anbi3i 1159 . 2 ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧))) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦))))
231, 22bitri 275 1 (𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cop 4631   × cxp 5682   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  m cmap 8867  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-ixp 8939
This theorem is referenced by:  funcf2lem2  48931  functhinc  49122
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