1stfcl - Metamath Proof Explorer

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Theorem 1stfcl 17104

Description: The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
1stfcl.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
1stfcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
1stfcl.d	⊢ (𝜑 → 𝐷 ∈ Cat)
1stfcl.p	⊢ 𝑃 = (𝐶 1^st_F 𝐷)

**Assertion**
Ref	Expression
1stfcl	⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶))

Proof of Theorem 1stfcl Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1stfcl.t	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		eqid 2764	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2764	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1, 2, 3	xpcbas 17085	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5		eqid 2764	. . . 4 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		1stfcl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		1stfcl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		1stfcl.p	. . . 4 ⊢ 𝑃 = (𝐶 1^st_F 𝐷)
9	1, 4, 5, 6, 7, 8	1stfval 17098	. . 3 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
10		fo1st 7385	. . . . . . . 8 ⊢ 1^st :V–onto→V
11		fofun 6298	. . . . . . . 8 ⊢ (1^st :V–onto→V → Fun 1^st )
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ Fun 1^st
13		fvex 6387	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
14		fvex 6387	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
15	13, 14	xpex 7159	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16		resfunexg 6671	. . . . . . 7 ⊢ ((Fun 1^st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
17	12, 15, 16	mp2an 683	. . . . . 6 ⊢ (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
18	15, 15	mpt2ex 7447	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
19	17, 18	op2ndd 7376	. . . . 5 ⊢ (𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩ → (2^nd ‘𝑃) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))))
20	9, 19	syl 17	. . . 4 ⊢ (𝜑 → (2^nd ‘𝑃) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))))
21	20	opeq2d 4565	. . 3 ⊢ (𝜑 → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩ = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
22	9, 21	eqtr4d 2801	. 2 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩)
23		eqid 2764	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2764	. . . 4 ⊢ (Id‘𝑇) = (Id‘𝑇)
25		eqid 2764	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
26		eqid 2764	. . . 4 ⊢ (comp‘𝑇) = (comp‘𝑇)
27		eqid 2764	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
28	1, 6, 7	xpccat 17097	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
29		f1stres 7389	. . . . 5 ⊢ (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐶)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐶))
31		eqid 2764	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))
32		ovex 6873	. . . . . . 7 ⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V
33		resfunexg 6671	. . . . . . 7 ⊢ ((Fun 1^st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
34	12, 32, 33	mp2an 683	. . . . . 6 ⊢ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
35	31, 34	fnmpt2i 7439	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))
36	20	fneq1d 6158	. . . . 5 ⊢ (𝜑 → ((2^nd ‘𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))))
37	35, 36	mpbiri 249	. . . 4 ⊢ (𝜑 → (2^nd ‘𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38		f1stres 7389	. . . . . 6 ⊢ (1^st ↾ (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))):(((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))⟶((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))
39	6	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
40	7	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41		simprl 787	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42		simprr 789	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
43	1, 4, 5, 39, 40, 8, 41, 42	1stf2 17100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2^nd ‘𝑃)𝑦) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))
44		eqid 2764	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	1, 4, 23, 44, 5, 41, 42	xpchom 17087	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(Hom ‘𝑇)𝑦) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))))
46	45	reseq2d 5564	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) = (1^st ↾ (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))))
47	43, 46	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2^nd ‘𝑃)𝑦) = (1^st ↾ (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))))
48	47	feq1d 6207	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2^nd ‘𝑃)𝑦):(((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))⟶((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) ↔ (1^st ↾ (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))):(((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))⟶((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))))
49	38, 48	mpbiri 249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2^nd ‘𝑃)𝑦):(((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))⟶((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)))
50		fvres 6393	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
51	50	ad2antrl 719	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
52		fvres 6393	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
53	52	ad2antll 720	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
54	51, 53	oveq12d 6859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) = ((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)))
55	45, 54	feq23d 6217	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2^nd ‘𝑃)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) ↔ (𝑥(2^nd ‘𝑃)𝑦):(((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))⟶((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))))
56	49, 55	mpbird 248	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2^nd ‘𝑃)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)))
57	28	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat)
58		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
59	4, 5, 24, 57, 58	catidcl 16609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60		fvres 6393	. . . . . . 7 ⊢ (((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1^st ‘((Id‘𝑇)‘𝑥)))
61	59, 60	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1^st ‘((Id‘𝑇)‘𝑥)))
62		1st2nd2 7404	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	62	adantl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
64	63	fveq2d 6378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
65	6	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
66	7	adantr 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
67		eqid 2764	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
68		xp1st 7397	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑥) ∈ (Base‘𝐶))
69	68	adantl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘𝑥) ∈ (Base‘𝐶))
70		xp2nd 7398	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
71	70	adantl 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
72	1, 65, 66, 2, 3, 25, 67, 24, 69, 71	xpcid 17096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩)
73	64, 72	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩)
74		fvex 6387	. . . . . . . 8 ⊢ ((Id‘𝐶)‘(1^st ‘𝑥)) ∈ V
75		fvex 6387	. . . . . . . 8 ⊢ ((Id‘𝐷)‘(2^nd ‘𝑥)) ∈ V
76	74, 75	op1std 7375	. . . . . . 7 ⊢ (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩ → (1^st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
77	73, 76	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
78	61, 77	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
79	1, 4, 5, 65, 66, 8, 58, 58	1stf2 17100	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2^nd ‘𝑃)𝑥) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑥)))
80	79	fveq1d 6376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2^nd ‘𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
81	50	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
82	81	fveq2d 6378	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐶)‘((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
83	78, 80, 82	3eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2^nd ‘𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
84	28	3ad2ant1 1163	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑇 ∈ Cat)
85		simp21 1263	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
86		simp22 1264	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
87		simp23 1265	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷)))
88		simp3l 1258	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦))
89		simp3r 1259	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))
90	4, 5, 26, 84, 85, 86, 87, 88, 89	catcocl 16612	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧))
91		fvres 6393	. . . . . . 7 ⊢ ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓) ∈ (𝑥(Hom ‘𝑇)𝑧) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (1^st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
92	90, 91	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (1^st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
93	1, 4, 5, 26, 85, 86, 87, 88, 89, 27	xpcco1st 17091	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (1^st ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘𝑥), (1^st ‘𝑦)⟩(comp‘𝐶)(1^st ‘𝑧))(1^st ‘𝑓)))
94	92, 93	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘𝑥), (1^st ‘𝑦)⟩(comp‘𝐶)(1^st ‘𝑧))(1^st ‘𝑓)))
95	6	3ad2ant1 1163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐶 ∈ Cat)
96	7	3ad2ant1 1163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → 𝐷 ∈ Cat)
97	1, 4, 5, 95, 96, 8, 85, 87	1stf2 17100	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2^nd ‘𝑃)𝑧) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑧)))
98	97	fveq1d 6376	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2^nd ‘𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((1^st ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
99	85, 50	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
100	86, 52	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
101	99, 100	opeq12d 4566	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ⟨((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩ = ⟨(1^st ‘𝑥), (1^st ‘𝑦)⟩)
102		fvres 6393	. . . . . . . 8 ⊢ (𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (1^st ‘𝑧))
103	87, 102	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (1^st ‘𝑧))
104	101, 103	oveq12d 6859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (⟨((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧)) = (⟨(1^st ‘𝑥), (1^st ‘𝑦)⟩(comp‘𝐶)(1^st ‘𝑧)))
105	1, 4, 5, 95, 96, 8, 86, 87	1stf2 17100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑦(2^nd ‘𝑃)𝑧) = (1^st ↾ (𝑦(Hom ‘𝑇)𝑧)))
106	105	fveq1d 6376	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2^nd ‘𝑃)𝑧)‘𝑔) = ((1^st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔))
107		fvres 6393	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧) → ((1^st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (1^st ‘𝑔))
108	89, 107	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (1^st ‘𝑔))
109	106, 108	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑦(2^nd ‘𝑃)𝑧)‘𝑔) = (1^st ‘𝑔))
110	1, 4, 5, 95, 96, 8, 85, 86	1stf2 17100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (𝑥(2^nd ‘𝑃)𝑦) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)))
111	110	fveq1d 6376	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2^nd ‘𝑃)𝑦)‘𝑓) = ((1^st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓))
112		fvres 6393	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (1^st ‘𝑓))
113	88, 112	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (1^st ‘𝑓))
114	111, 113	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2^nd ‘𝑃)𝑦)‘𝑓) = (1^st ‘𝑓))
115	104, 109, 114	oveq123d 6862	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → (((𝑦(2^nd ‘𝑃)𝑧)‘𝑔)(⟨((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2^nd ‘𝑃)𝑦)‘𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘𝑥), (1^st ‘𝑦)⟩(comp‘𝐶)(1^st ‘𝑧))(1^st ‘𝑓)))
116	94, 98, 115	3eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑧 ∈ ((Base‘𝐶) × (Base‘𝐷))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))) → ((𝑥(2^nd ‘𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (((𝑦(2^nd ‘𝑃)𝑧)‘𝑔)(⟨((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐶)((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2^nd ‘𝑃)𝑦)‘𝑓)))
117	4, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 83, 116	isfuncd 16791	. . 3 ⊢ (𝜑 → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2^nd ‘𝑃))
118		df-br 4809	. . 3 ⊢ ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2^nd ‘𝑃) ↔ ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩ ∈ (𝑇 Func 𝐶))
119	117, 118	sylib 209	. 2 ⊢ (𝜑 → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩ ∈ (𝑇 Func 𝐶))
120	22, 119	eqeltrd 2843	1 ⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1107 = wceq 1652 ∈ wcel 2155 Vcvv 3349 ⟨cop 4339 class class class wbr 4808 × cxp 5274 ↾ cres 5278 Fun wfun 6061 Fn wfn 6062 ⟶wf 6063 –onto→wfo 6065 ‘cfv 6067 (class class class)co 6841 ↦ cmpt2 6843 1^st c1st 7363 2^nd c2nd 7364 Basecbs 16131 Hom chom 16226 compcco 16227 Catccat 16591 Idccid 16592 Func cfunc 16780 ×_c cxpc 17075 1^st_F c1stf 17076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2069 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2349 ax-ext 2742 ax-rep 4929 ax-sep 4940 ax-nul 4948 ax-pow 5000 ax-pr 5061 ax-un 7146 ax-cnex 10244 ax-resscn 10245 ax-1cn 10246 ax-icn 10247 ax-addcl 10248 ax-addrcl 10249 ax-mulcl 10250 ax-mulrcl 10251 ax-mulcom 10252 ax-addass 10253 ax-mulass 10254 ax-distr 10255 ax-i2m1 10256 ax-1ne0 10257 ax-1rid 10258 ax-rnegex 10259 ax-rrecex 10260 ax-cnre 10261 ax-pre-lttri 10262 ax-pre-lttrn 10263 ax-pre-ltadd 10264 ax-pre-mulgt0 10265

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2062 df-mo 2564 df-eu 2581 df-clab 2751 df-cleq 2757 df-clel 2760 df-nfc 2895 df-ne 2937 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3351 df-sbc 3596 df-csb 3691 df-dif 3734 df-un 3736 df-in 3738 df-ss 3745 df-pss 3747 df-nul 4079 df-if 4243 df-pw 4316 df-sn 4334 df-pr 4336 df-tp 4338 df-op 4340 df-uni 4594 df-int 4633 df-iun 4677 df-br 4809 df-opab 4871 df-mpt 4888 df-tr 4911 df-id 5184 df-eprel 5189 df-po 5197 df-so 5198 df-fr 5235 df-we 5237 df-xp 5282 df-rel 5283 df-cnv 5284 df-co 5285 df-dm 5286 df-rn 5287 df-res 5288 df-ima 5289 df-pred 5864 df-ord 5910 df-on 5911 df-lim 5912 df-suc 5913 df-iota 6030 df-fun 6069 df-fn 6070 df-f 6071 df-f1 6072 df-fo 6073 df-f1o 6074 df-fv 6075 df-riota 6802 df-ov 6844 df-oprab 6845 df-mpt2 6846 df-om 7263 df-1st 7365 df-2nd 7366 df-wrecs 7609 df-recs 7671 df-rdg 7709 df-1o 7763 df-oadd 7767 df-er 7946 df-map 8061 df-ixp 8113 df-en 8160 df-dom 8161 df-sdom 8162 df-fin 8163 df-pnf 10329 df-mnf 10330 df-xr 10331 df-ltxr 10332 df-le 10333 df-sub 10521 df-neg 10522 df-nn 11274 df-2 11334 df-3 11335 df-4 11336 df-5 11337 df-6 11338 df-7 11339 df-8 11340 df-9 11341 df-n0 11538 df-z 11624 df-dec 11740 df-uz 11886 df-fz 12533 df-struct 16133 df-ndx 16134 df-slot 16135 df-base 16137 df-hom 16239 df-cco 16240 df-cat 16595 df-cid 16596 df-func 16784 df-xpc 17079 df-1stf 17080

This theorem is referenced by: prf1st 17111 1st2ndprf 17113 uncfcl 17142 uncf1 17143 uncf2 17144 diagcl 17148 diag11 17150 diag12 17151 diag2 17152 yonedalem1 17179 yonedalem21 17180 yonedalem22 17185

W3C validator