1stfcl - Metamath Proof Explorer

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Theorem 1stfcl 17305

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 2778	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2778	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1, 2, 3	xpcbas 17286	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5		eqid 2778	. . . 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 17299	. . 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 7521	. . . . . . . 8 ⊢ 1^st :V–onto→V
11		fofun 6420	. . . . . . . 8 ⊢ (1^st :V–onto→V → Fun 1^st )
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ Fun 1^st
13		fvex 6512	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
14		fvex 6512	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
15	13, 14	xpex 7293	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16		resfunexg 6804	. . . . . . 7 ⊢ ((Fun 1^st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
17	12, 15, 16	mp2an 679	. . . . . 6 ⊢ (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
18	15, 15	mpoex 7585	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
19	17, 18	op2ndd 7512	. . . . 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 4684	. . 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 2817	. 2 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩)
23		eqid 2778	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2778	. . . 4 ⊢ (Id‘𝑇) = (Id‘𝑇)
25		eqid 2778	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
26		eqid 2778	. . . 4 ⊢ (comp‘𝑇) = (comp‘𝑇)
27		eqid 2778	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
28	1, 6, 7	xpccat 17298	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
29		f1stres 7525	. . . . 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 2778	. . . . . 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 7008	. . . . . . 7 ⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V
33		resfunexg 6804	. . . . . . 7 ⊢ ((Fun 1^st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
34	12, 32, 33	mp2an 679	. . . . . 6 ⊢ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
35	31, 34	fnmpoi 7576	. . . . 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 6279	. . . . 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 250	. . . 4 ⊢ (𝜑 → (2^nd ‘𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38		f1stres 7525	. . . . . 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 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
40	7	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41		simprl 758	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42		simprr 760	. . . . . . . . 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 17301	. . . . . . . 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 2778	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	1, 4, 23, 44, 5, 41, 42	xpchom 17288	. . . . . . . . 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 5695	. . . . . . . 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 2814	. . . . . . 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 6329	. . . . . 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 250	. . . . 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 6518	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
51	50	ad2antrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
52		fvres 6518	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
53	52	ad2antll 716	. . . . . . 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 6994	. . . . . 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 6339	. . . . 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 249	. . . 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 473	. . . . . . . 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 16811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60	59	fvresd 6519	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1^st ‘((Id‘𝑇)‘𝑥)))
61		1st2nd2 7540	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
62	61	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	62	fveq2d 6503	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
64	6	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
65	7	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
66		eqid 2778	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
67		xp1st 7533	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑥) ∈ (Base‘𝐶))
68	67	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘𝑥) ∈ (Base‘𝐶))
69		xp2nd 7534	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
70	69	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
71	1, 64, 65, 2, 3, 25, 66, 24, 68, 70	xpcid 17297	. . . . . . . 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))⟩)
72	63, 71	eqtrd 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩)
73		fvex 6512	. . . . . . . 8 ⊢ ((Id‘𝐶)‘(1^st ‘𝑥)) ∈ V
74		fvex 6512	. . . . . . . 8 ⊢ ((Id‘𝐷)‘(2^nd ‘𝑥)) ∈ V
75	73, 74	op1std 7511	. . . . . . 7 ⊢ (((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩ → (1^st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
76	72, 75	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
77	60, 76	eqtrd 2814	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
78	1, 4, 5, 64, 65, 8, 58, 58	1stf2 17301	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2^nd ‘𝑃)𝑥) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑥)))
79	78	fveq1d 6501	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2^nd ‘𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)))
80	50	adantl 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
81	80	fveq2d 6503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝐶)‘((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)) = ((Id‘𝐶)‘(1^st ‘𝑥)))
82	77, 79, 81	3eqtr4d 2824	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((𝑥(2^nd ‘𝑃)𝑥)‘((Id‘𝑇)‘𝑥)) = ((Id‘𝐶)‘((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥)))
83	28	3ad2ant1 1113	. . . . . . . 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 1186	. . . . . . . 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 1187	. . . . . . . 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 1188	. . . . . . . 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 1181	. . . . . . . 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 1182	. . . . . . . 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 16814	. . . . . . 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 6519	. . . . . 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)))
91	1, 4, 5, 26, 84, 85, 86, 87, 88, 27	xpcco1st 17292	. . . . . 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)))
92	90, 91	eqtrd 2814	. . . . 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)))
93	6	3ad2ant1 1113	. . . . . . 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 1113	. . . . . . 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	1stf2 17301	. . . . . 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)))
96	95	fveq1d 6501	. . . . 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)))
97	84	fvresd 6519	. . . . . . . 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))
98	85	fvresd 6519	. . . . . . . 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))
99	97, 98	opeq12d 4685	. . . . . . 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)⟩)
100	86	fvresd 6519	. . . . . . 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))
101	99, 100	oveq12d 6994	. . . . . 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)))
102	1, 4, 5, 93, 94, 8, 85, 86	1stf2 17301	. . . . . . . 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)))
103	102	fveq1d 6501	. . . . . . 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))
104	88	fvresd 6519	. . . . . . 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))
105	103, 104	eqtrd 2814	. . . . . 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))
106	1, 4, 5, 93, 94, 8, 84, 85	1stf2 17301	. . . . . . . 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)))
107	106	fveq1d 6501	. . . . . . 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))
108	87	fvresd 6519	. . . . . . 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))
109	107, 108	eqtrd 2814	. . . . . 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))
110	101, 105, 109	oveq123d 6997	. . . . 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)))
111	92, 96, 110	3eqtr4d 2824	. . . 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)))
112	4, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 82, 111	isfuncd 16993	. . 3 ⊢ (𝜑 → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2^nd ‘𝑃))
113		df-br 4930	. . 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))
114	112, 113	sylib 210	. 2 ⊢ (𝜑 → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩ ∈ (𝑇 Func 𝐶))
115	22, 114	eqeltrd 2866	1 ⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ⟨cop 4447 class class class wbr 4929 × cxp 5405 ↾ cres 5409 Fun wfun 6182 Fn wfn 6183 ⟶wf 6184 –onto→wfo 6186 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 1^st c1st 7499 2^nd c2nd 7500 Basecbs 16339 Hom chom 16432 compcco 16433 Catccat 16793 Idccid 16794 Func cfunc 16982 ×_c cxpc 17276 1^st_F c1stf 17277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-hom 16445 df-cco 16446 df-cat 16797 df-cid 16798 df-func 16986 df-xpc 17280 df-1stf 17281

This theorem is referenced by: prf1st 17312 1st2ndprf 17314 uncfcl 17343 uncf1 17344 uncf2 17345 diagcl 17349 diag11 17351 diag12 17352 diag2 17353 yonedalem1 17380 yonedalem21 17381 yonedalem22 17386

W3C validator