Theorem funcf2lem 47638

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 8895	. 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 6892	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
3		df-ov 7412	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2791	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
5		vex 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
6		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
7	5, 6	op1std 7985	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
8	7	fveq2d 6896	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
9	5, 6	op2ndd 7986	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
10	9	fveq2d 6896	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
11	8, 10	oveq12d 7427	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
12		fveq2 6892	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
13		df-ov 7412	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
14	12, 13	eqtr4di 2791	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
15	11, 14	oveq12d 7427	. . . . . 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 2828	. . . . 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 7442	. . . . . 6 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
18		ovex 7442	. . . . . 6 ⊢ (𝑥𝐻𝑦) ∈ V
19	17, 18	elmap 8865	. . . . 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 5842	. . 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 1160	. 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 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  ⟨cop 4635   × cxp 5675   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  1^st c1st 7973  2^nd c2nd 7974   ↑_m cmap 8820  Xcixp 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-ixp 8892

This theorem is referenced by:  functhinc  47665