Theorem isfuncd 18033

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 6897	. . . . 5 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7765	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
6		fnex 7221	. . . 4 ⊢ ((𝐺 Fn (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ∈ V) → 𝐺 ∈ V)
7	2, 5, 6	sylancl 598	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
8		isfuncd.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
9		ovex 7451	. . . . . . 7 ⊢ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∈ V
10		ovex 7451	. . . . . . 7 ⊢ (𝑥𝐻𝑦) ∈ V
11	9, 10	elmap 8892	. . . . . 6 ⊢ ((𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
12	8, 11	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
13	12	ralrimivva 3206	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦) ∈ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↑m (𝑥𝐻𝑦)))
14		fveq2 6883	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
15		df-ov 7421	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
16	14, 15	eqtr4di 2814	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
17		vex 3455	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
18		vex 3455	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
19	17, 18	op1std 8009	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
20	19	fveq2d 6887	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
21	17, 18	op2ndd 8010	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
22	21	fveq2d 6887	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd ‘𝑧)) = (𝐹‘𝑦))
23	20, 22	oveq12d 7436	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
24		fveq2 6883	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
25		df-ov 7421	. . . . . . . 8 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
26	24, 25	eqtr4di 2814	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
27	23, 26	oveq12d 7436	. . . . . 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 2855	. . . . 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 5818	. . . 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 237	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺‘𝑧) ∈ (((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
31		elixp2 8922	. . 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 1362	. 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 1139	. . . . . . . 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 1373	. . . . . . 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 433	. . . . . 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 3204	. . . . 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 3206	. . . 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 521	. . 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 3155	. 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 18032	. 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 1361	1 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⟨cop 4590   class class class wbr 5103   × cxp 5649   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998   ↑_m cmap 8840  Xcixp 8918  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  Idccid 17832   Func cfunc 18022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-map 8842  df-ixp 8919  df-func 18026

This theorem is used by:  funcoppc  18043  funcres  18064  catcisolem  18278  funcestrcsetc  18316  funcsetcestrc  18331  1stfcl  18364  2ndfcl  18365  prfcl  18370  evlfcl  18389  curf1cl  18395  curfcl  18399  hofcl  18426  funcringcsetcALTV2  49365  funcringcsetcALTV  49388  swapffunc  50359  fucofunc  50436  fucoppc  50487