Theorem funcf2lem 49013

Description: A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)

Assertion

Ref	Expression
funcf2lem	⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))

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

Step	Hyp	Ref	Expression
1		elixp2 8920	. 2 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))))
2		fveq2 6881	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
3		df-ov 7413	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2789	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
5		vex 3468	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6		vex 3468	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
7	5, 6	op1std 8003	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
8	7	fveq2d 6885	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
9	5, 6	op2ndd 8004	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
10	9	fveq2d 6885	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
11	8, 10	oveq12d 7428	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
12		fveq2 6881	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
13		df-ov 7413	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
14	12, 13	eqtr4di 2789	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
15	11, 14	oveq12d 7428	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
16	4, 15	eleq12d 2829	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦))))
17		ovex 7443	. . . . . 6 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
18		ovex 7443	. . . . . 6 ⊢ (𝑥𝐻𝑦) ∈ V
19	17, 18	elmap 8890	. . . . 5 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
20	16, 19	bitrdi 287	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
21	20	ralxp 5826	. . 3 ⊢ (∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
22	21	3anbi3i 1159	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧))) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
23	1, 22	bitri 275	1 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))

Syntax hints:   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464  ⟨cop 4612   × cxp 5657   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992   ↑_m cmap 8845  Xcixp 8916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-map 8847  df-ixp 8917

This theorem is referenced by:  funcf2lem2  49014  functhinc  49301