1stfcl - Metamath Proof Explorer

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Theorem 1stfcl 18154

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 2739	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2739	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1, 2, 3	xpcbas 18135	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5		eqid 2739	. . . 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 18148	. . 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 7951	. . . . . . . 8 ⊢ 1^st :V–onto→V
11		fofun 6740	. . . . . . . 8 ⊢ (1^st :V–onto→V → Fun 1^st )
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ Fun 1^st
13		fvex 6840	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
14		fvex 6840	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
15	13, 14	xpex 7696	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16		resfunexg 7159	. . . . . . 7 ⊢ ((Fun 1^st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
17	12, 15, 16	mp2an 698	. . . . . 6 ⊢ (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
18	15, 15	mpoex 8021	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
19	17, 18	op2ndd 7942	. . . . 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 4811	. . 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 2777	. 2 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩)
23		eqid 2739	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2739	. . . 4 ⊢ (Id‘𝑇) = (Id‘𝑇)
25		eqid 2739	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
26		eqid 2739	. . . 4 ⊢ (comp‘𝑇) = (comp‘𝑇)
27		eqid 2739	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
28	1, 6, 7	xpccat 18147	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
29		f1stres 7955	. . . . 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 2739	. . . . . 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 7389	. . . . . . 7 ⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V
33		resfunexg 7159	. . . . . . 7 ⊢ ((Fun 1^st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
34	12, 32, 33	mp2an 698	. . . . . 6 ⊢ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
35	31, 34	fnmpoi 8012	. . . . 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 6578	. . . . 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 259	. . . 4 ⊢ (𝜑 → (2^nd ‘𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38		f1stres 7955	. . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐶 ∈ Cat)
40	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝐷 ∈ Cat)
41		simprl 776	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42		simprr 778	. . . . . . . . 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 18150	. . . . . . . 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 2739	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	1, 4, 23, 44, 5, 41, 42	xpchom 18137	. . . . . . . . 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 5931	. . . . . . . 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 2774	. . . . . . 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 6637	. . . . . 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 259	. . . . 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 6846	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
51	50	ad2antrl 734	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
52		fvres 6846	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
53	52	ad2antll 735	. . . . . . 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 7374	. . . . . 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 6650	. . . . 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 258	. . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑇 ∈ Cat)
58		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
59	4, 5, 24, 57, 58	catidcl 17639	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60	59	fvresd 6847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1^st ‘((Id‘𝑇)‘𝑥)))
61		1st2nd2 7970	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
62	61	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	62	fveq2d 6831	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩))
64	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐶 ∈ Cat)
65	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝐷 ∈ Cat)
66		eqid 2739	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
67		xp1st 7963	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑥) ∈ (Base‘𝐶))
68	67	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘𝑥) ∈ (Base‘𝐶))
69		xp2nd 7964	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
70	69	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
71	1, 64, 65, 2, 3, 25, 66, 24, 68, 70	xpcid 18146	. . . . . . . 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 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩)
73		fvex 6840	. . . . . . . 8 ⊢ ((Id‘𝐶)‘(1^st ‘𝑥)) ∈ V
74		fvex 6840	. . . . . . . 8 ⊢ ((Id‘𝐷)‘(2^nd ‘𝑥)) ∈ V
75	73, 74	op1std 7941	. . . . . . 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 2774	. . . . 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 18150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2^nd ‘𝑃)𝑥) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑥)))
79	78	fveq1d 6829	. . . . 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 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
81	80	fveq2d 6831	. . . . 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 2784	. . . 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 1139	. . . . . . . 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 1213	. . . . . . . 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 1214	. . . . . . . 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 1215	. . . . . . . 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 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))) → 𝑓 ∈ (𝑥(Hom ‘𝑇)𝑦))
88		simp3r 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))) → 𝑔 ∈ (𝑦(Hom ‘𝑇)𝑧))
89	4, 5, 26, 83, 84, 85, 86, 87, 88	catcocl 17642	. . . . . . 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 6847	. . . . . 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 18141	. . . . . 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 2774	. . . . 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 1139	. . . . . . 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 1139	. . . . . . 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 18150	. . . . . 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 6829	. . . . 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 6847	. . . . . . . 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 6847	. . . . . . . 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 4812	. . . . . . 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 6847	. . . . . . 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 7374	. . . . . 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 18150	. . . . . . . 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 6829	. . . . . . 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 6847	. . . . . . 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 2774	. . . . . 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 18150	. . . . . . . 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 6829	. . . . . . 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 6847	. . . . . . 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 2774	. . . . . 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 7377	. . . . 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 2784	. . . 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 17823	. . 3 ⊢ (𝜑 → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2^nd ‘𝑃))
113		df-br 5073	. . 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 219	. 2 ⊢ (𝜑 → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩ ∈ (𝑇 Func 𝐶))
115	22, 114	eqeltrd 2839	1 ⊢ (𝜑 → 𝑃 ∈ (𝑇 Func 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ⟨cop 4561 class class class wbr 5072 × cxp 5616 ↾ cres 5620 Fun wfun 6479 Fn wfn 6480 ⟶wf 6481 –onto→wfo 6483 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17170 Hom chom 17222 compcco 17223 Catccat 17621 Idccid 17622 Func cfunc 17812 ×_c cxpc 18125 1^st_F c1stf 18126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-hom 17235 df-cco 17236 df-cat 17625 df-cid 17626 df-func 17816 df-xpc 18129 df-1stf 18130

This theorem is referenced by: prf1st 18161 1st2ndprf 18163 uncfcl 18192 uncf1 18193 uncf2 18194 diagcl 18198 diag11 18200 diag12 18201 diag2 18202 yonedalem1 18229 yonedalem21 18230 yonedalem22 18235 oppc1stf 49778 diagpropd 49782

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