Theorem 2ndfcl 18121

Description: The second projection functor is a functor onto the right 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)
2ndfcl.p	⊢ 𝑄 = (𝐶 2ndF 𝐷)

Assertion

Ref	Expression
2ndfcl	⊢ (𝜑 → 𝑄 ∈ (𝑇 Func 𝐷))

Proof of Theorem 2ndfcl

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 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2736	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1, 2, 3	xpcbas 18101	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5		eqid 2736	. . . 4 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
6		1stfcl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		1stfcl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		2ndfcl.p	. . . 4 ⊢ 𝑄 = (𝐶 2ndF 𝐷)
9	1, 4, 5, 6, 7, 8	2ndfval 18117	. . 3 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
10		fo2nd 7954	. . . . . . . 8 ⊢ 2nd :V–onto→V
11		fofun 6747	. . . . . . . 8 ⊢ (2nd :V–onto→V → Fun 2nd )
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ Fun 2nd
13		fvex 6847	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
14		fvex 6847	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
15	13, 14	xpex 7698	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16		resfunexg 7161	. . . . . . 7 ⊢ ((Fun 2nd ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
17	12, 15, 16	mp2an 692	. . . . . 6 ⊢ (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
18	15, 15	mpoex 8023	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
19	17, 18	op2ndd 7944	. . . . 5 ⊢ (𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩ → (2nd ‘𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))))
20	9, 19	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘𝑄) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))))
21	20	opeq2d 4836	. . 3 ⊢ (𝜑 → ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd ‘𝑄)⟩ = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))⟩)
22	9, 21	eqtr4d 2774	. 2 ⊢ (𝜑 → 𝑄 = ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd ‘𝑄)⟩)
23		eqid 2736	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
24		eqid 2736	. . . 4 ⊢ (Id‘𝑇) = (Id‘𝑇)
25		eqid 2736	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
26		eqid 2736	. . . 4 ⊢ (comp‘𝑇) = (comp‘𝑇)
27		eqid 2736	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
28	1, 6, 7	xpccat 18113	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
29		f2ndres 7958	. . . . 5 ⊢ (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷))):((Base‘𝐶) × (Base‘𝐷))⟶(Base‘𝐷))
31		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
32		ovex 7391	. . . . . . 7 ⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V
33		resfunexg 7161	. . . . . . 7 ⊢ ((Fun 2nd ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
34	12, 32, 33	mp2an 692	. . . . . 6 ⊢ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
35	31, 34	fnmpoi 8014	. . . . 5 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))
36	20	fneq1d 6585	. . . . 5 ⊢ (𝜑 → ((2nd ‘𝑄) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))) ↔ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (2nd ↾ (𝑥(Hom ‘𝑇)𝑦))) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷)))))
37	35, 36	mpbiri 258	. . . 4 ⊢ (𝜑 → (2nd ‘𝑄) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38		f2ndres 7958	. . . . . 6 ⊢ (2nd ↾ (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))
39	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
40	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42		simprr 772	. . . . . . . . 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	2ndf2 18119	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
44		eqid 2736	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
45	1, 4, 44, 23, 5, 41, 42	xpchom 18103	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(Hom ‘𝑇)𝑦) = (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))
46	45	reseq2d 5938	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)) = (2nd ↾ (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))))
47	43, 46	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦) = (2nd ↾ (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))))
48	47	feq1d 6644	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)) ↔ (2nd ↾ (((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))
49	38, 48	mpbiri 258	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))
50		fvres 6853	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd ‘𝑥))
51	50	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd ‘𝑥))
52		fvres 6853	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd ‘𝑦))
53	52	ad2antll 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd ‘𝑦))
54	51, 53	oveq12d 7376	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) = ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))
55	45, 54	feq23d 6657	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((𝑥(2nd ‘𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)) ↔ (𝑥(2nd ‘𝑄)𝑦):(((1st ‘𝑥)(Hom ‘𝐶)(1st ‘𝑦)) × ((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦)))⟶((2nd ‘𝑥)(Hom ‘𝐷)(2nd ‘𝑦))))
56	49, 55	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → (𝑥(2nd ‘𝑄)𝑦):(𝑥(Hom ‘𝑇)𝑦)⟶(((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)(Hom ‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)))
57	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat)
58		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
59	4, 5, 24, 57, 58	catidcl 17605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60	59	fvresd 6854	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (2nd ‘((Id‘𝑇)‘𝑥)))
61		1st2nd2 7972	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
62	61	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
63	62	fveq2d 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
64	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
65	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
66		eqid 2736	. . . . . . . . 9 ⊢ (Id‘𝐶) = (Id‘𝐶)
67		xp1st 7965	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑥) ∈ (Base‘𝐶))
68	67	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1st ‘𝑥) ∈ (Base‘𝐶))
69		xp2nd 7966	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd ‘𝑥) ∈ (Base‘𝐷))
70	69	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd ‘𝑥) ∈ (Base‘𝐷))
71	1, 64, 65, 2, 3, 66, 25, 24, 68, 70	xpcid 18112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))⟩)
72	63, 71	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))⟩)
73		fvex 6847	. . . . . . . 8 ⊢ ((Id‘𝐶)‘(1st ‘𝑥)) ∈ V
74		fvex 6847	. . . . . . . 8 ⊢ ((Id‘𝐷)‘(2nd ‘𝑥)) ∈ V
75	73, 74	op2ndd 7944	. . . . . . 7 ⊢ (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1st ‘𝑥)), ((Id‘𝐷)‘(2nd ‘𝑥))⟩ → (2nd ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥)))
76	72, 75	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2nd ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥)))
77	60, 76	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥)))
78	1, 4, 5, 64, 65, 8, 58, 58	2ndf2 18119	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2nd ‘𝑄)𝑥) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑥)))
79	78	fveq1d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd ‘𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
80	50	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd ‘𝑥))
81	80	fveq2d 6838	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐷)‘((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐷)‘(2nd ‘𝑥)))
82	77, 79, 81	3eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2nd ‘𝑄)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐷)‘((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
83	28	3ad2ant1 1133	. . . . . . . 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)
84		simp21 1207	. . . . . . . 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)))
85		simp22 1208	. . . . . . . 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)))
86		simp23 1209	. . . . . . . 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)))
87		simp3l 1202	. . . . . . . 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))
88		simp3r 1203	. . . . . . . 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))
89	4, 5, 26, 83, 84, 85, 86, 87, 88	catcocl 17608	. . . . . . 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))
90	89	fvresd 6854	. . . . . 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))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (2nd ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
91	1, 4, 5, 26, 84, 85, 86, 87, 88, 27	xpcco2nd 18108	. . . . . 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))) → (2nd ‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓)))
92	90, 91	eqtrd 2771	. . . . 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))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓)))
93	6	3ad2ant1 1133	. . . . . . 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)
94	7	3ad2ant1 1133	. . . . . . 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)
95	1, 4, 5, 93, 94, 8, 84, 86	2ndf2 18119	. . . . . 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(2nd ‘𝑄)𝑧) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑧)))
96	95	fveq1d 6836	. . . . 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(2nd ‘𝑄)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑧))‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)))
97	84	fvresd 6854	. . . . . . . 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))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (2nd ‘𝑥))
98	85	fvresd 6854	. . . . . . . 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))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (2nd ‘𝑦))
99	97, 98	opeq12d 4837	. . . . . . 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))) → ⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩ = ⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩)
100	86	fvresd 6854	. . . . . . 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))) → ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧) = (2nd ‘𝑧))
101	99, 100	oveq12d 7376	. . . . . 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))) → (⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧)) = (⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐷)(2nd ‘𝑧)))
102	1, 4, 5, 93, 94, 8, 85, 86	2ndf2 18119	. . . . . . . 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(2nd ‘𝑄)𝑧) = (2nd ↾ (𝑦(Hom ‘𝑇)𝑧)))
103	102	fveq1d 6836	. . . . . . 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(2nd ‘𝑄)𝑧)‘𝑔) = ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔))
104	88	fvresd 6854	. . . . . . 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))) → ((2nd ↾ (𝑦(Hom ‘𝑇)𝑧))‘𝑔) = (2nd ‘𝑔))
105	103, 104	eqtrd 2771	. . . . . 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(2nd ‘𝑄)𝑧)‘𝑔) = (2nd ‘𝑔))
106	1, 4, 5, 93, 94, 8, 84, 85	2ndf2 18119	. . . . . . . 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(2nd ‘𝑄)𝑦) = (2nd ↾ (𝑥(Hom ‘𝑇)𝑦)))
107	106	fveq1d 6836	. . . . . . 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(2nd ‘𝑄)𝑦)‘𝑓) = ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓))
108	87	fvresd 6854	. . . . . . 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))) → ((2nd ↾ (𝑥(Hom ‘𝑇)𝑦))‘𝑓) = (2nd ‘𝑓))
109	107, 108	eqtrd 2771	. . . . . 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(2nd ‘𝑄)𝑦)‘𝑓) = (2nd ‘𝑓))
110	101, 105, 109	oveq123d 7379	. . . . 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(2nd ‘𝑄)𝑧)‘𝑔)(⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd ‘𝑄)𝑦)‘𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘𝑥), (2nd ‘𝑦)⟩(comp‘𝐷)(2nd ‘𝑧))(2nd ‘𝑓)))
111	92, 96, 110	3eqtr4d 2781	. . . 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(2nd ‘𝑄)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑇)𝑧)𝑓)) = (((𝑦(2nd ‘𝑄)𝑧)‘𝑔)(⟨((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥), ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦)⟩(comp‘𝐷)((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑧))((𝑥(2nd ‘𝑄)𝑦)‘𝑓)))
112	4, 3, 5, 23, 24, 25, 26, 27, 28, 7, 30, 37, 56, 82, 111	isfuncd 17789	. . 3 ⊢ (𝜑 → (2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐷)(2nd ‘𝑄))
113		df-br 5099	. . 3 ⊢ ((2nd ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐷)(2nd ‘𝑄) ↔ ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd ‘𝑄)⟩ ∈ (𝑇 Func 𝐷))
114	112, 113	sylib 218	. 2 ⊢ (𝜑 → ⟨(2nd ↾ ((Base‘𝐶) × (Base‘𝐷))), (2nd ‘𝑄)⟩ ∈ (𝑇 Func 𝐷))
115	22, 114	eqeltrd 2836	1 ⊢ (𝜑 → 𝑄 ∈ (𝑇 Func 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   class class class wbr 5098   × cxp 5622   ↾ cres 5626  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Hom chom 17188  compcco 17189  Catccat 17587  Idccid 17588   Func cfunc 17778   ×_c cxpc 18091   2^nd_F c2ndf 18093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-xpc 18095  df-2ndf 18097

This theorem is referenced by:  prf2nd  18128  1st2ndprf  18129  uncfcl  18158  uncf1  18159  uncf2  18160  curf2ndf  18170  yonedalem1  18195  yonedalem21  18196  yonedalem22  18201  oppc2ndf  49530