Theorem isfuncd 17129

Description: Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
isfunc.b	⊢ 𝐵 = (Base‘𝐷)
isfunc.c	⊢ 𝐶 = (Base‘𝐸)
isfunc.h	⊢ 𝐻 = (Hom ‘𝐷)
isfunc.j	⊢ 𝐽 = (Hom ‘𝐸)
isfunc.1	⊢ 1 = (Id‘𝐷)
isfunc.i	⊢ 𝐼 = (Id‘𝐸)
isfunc.x	⊢ · = (comp‘𝐷)
isfunc.o	⊢ 𝑂 = (comp‘𝐸)
isfunc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
isfunc.e	⊢ (𝜑 → 𝐸 ∈ Cat)
isfuncd.1	⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
isfuncd.2	⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
isfuncd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
isfuncd.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
isfuncd.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))

Assertion

Ref	Expression
isfuncd	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

Distinct variable groups:   𝑚,𝑛,𝑥,𝑦,𝑧,𝐵   𝐷,𝑚,𝑛,𝑥,𝑦,𝑧   𝑚,𝐸,𝑛,𝑥,𝑦,𝑧   𝑚,𝐻,𝑛,𝑥,𝑦,𝑧   𝑚,𝐹,𝑛,𝑥,𝑦,𝑧   𝑚,𝐺,𝑛,𝑥,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝜑,𝑚,𝑛,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑚,𝑛)   · (𝑥,𝑦,𝑧,𝑚,𝑛)   1 (𝑥,𝑦,𝑧,𝑚,𝑛)   𝐼(𝑥,𝑦,𝑧,𝑚,𝑛)   𝐽(𝑚,𝑛)   𝑂(𝑥,𝑦,𝑧,𝑚,𝑛)

Proof of Theorem isfuncd

Step	Hyp	Ref	Expression
1		isfuncd.1	. 2 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
2		isfuncd.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
3		isfunc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
4	3	fvexi 6678	. . . . 5 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7470	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
6		fnex 6974	. . . 4 ⊢ ((𝐺 Fn (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ∈ V) → 𝐺 ∈ V)
7	2, 5, 6	sylancl 588	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
8		isfuncd.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
9		ovex 7183	. . . . . . 7 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
10		ovex 7183	. . . . . . 7 ⊢ (𝑥𝐻𝑦) ∈ V
11	9, 10	elmap 8429	. . . . . 6 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
12	8, 11	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
13	12	ralrimivva 3191	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
14		fveq2 6664	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
15		df-ov 7153	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
16	14, 15	syl6eqr 2874	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
17		vex 3497	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3497	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	op1std 7693	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
20	19	fveq2d 6668	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
21	17, 18	op2ndd 7694	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
22	21	fveq2d 6668	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
23	20, 22	oveq12d 7168	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
24		fveq2 6664	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
25		df-ov 7153	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
26	24, 25	syl6eqr 2874	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
27	23, 26	oveq12d 7168	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
28	16, 27	eleq12d 2907	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦))))
29	28	ralxp 5706	. . . 4 ⊢ (∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
30	13, 29	sylibr 236	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
31		elixp2 8459	. . 3 ⊢ (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))))
32	7, 2, 30, 31	syl3anbrc 1339	. 2 ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
33		isfuncd.4	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
34		isfuncd.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))
35	34	3expia 1117	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
36	35	3exp2 1350	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐵 → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))))
37	36	imp43 430	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
38	37	ralrimivv 3190	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))
39	38	ralrimivva 3191	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))
40	33, 39	jca 514	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
41	40	ralrimiva 3182	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))
42		isfunc.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
43		isfunc.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
44		isfunc.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
45		isfunc.1	. . 3 ⊢ 1 = (Id‘𝐷)
46		isfunc.i	. . 3 ⊢ 𝐼 = (Id‘𝐸)
47		isfunc.x	. . 3 ⊢ · = (comp‘𝐷)
48		isfunc.o	. . 3 ⊢ 𝑂 = (comp‘𝐸)
49		isfunc.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
50		isfunc.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
51	3, 42, 43, 44, 45, 46, 47, 48, 49, 50	isfunc 17128	. 2 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘( 1 ‘𝑥)) = (𝐼‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩𝑂(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
52	1, 32, 41, 51	mpbir3and 1338	1 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ⟨cop 4566   class class class wbr 5058   × cxp 5547   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682   ↑_m cmap 8400  Xcixp 8455  Basecbs 16477  Hom chom 16570  compcco 16571  Catccat 16929  Idccid 16930   Func cfunc 17118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-ixp 8456  df-func 17122

This theorem is referenced by:  funcoppc  17139  funcres  17160  catcisolem  17360  funcestrcsetc  17393  funcsetcestrc  17408  1stfcl  17441  2ndfcl  17442  prfcl  17447  evlfcl  17466  curf1cl  17472  curfcl  17476  hofcl  17503  funcringcsetcALTV2  44310  funcringcsetcALTV  44333