1stfcl - Metamath Proof Explorer

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Theorem 1stfcl 17830

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 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2738	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
4	1, 2, 3	xpcbas 17811	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
5		eqid 2738	. . . 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 17824	. . 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 7824	. . . . . . . 8 ⊢ 1^st :V–onto→V
11		fofun 6673	. . . . . . . 8 ⊢ (1^st :V–onto→V → Fun 1^st )
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ Fun 1^st
13		fvex 6769	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
14		fvex 6769	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
15	13, 14	xpex 7581	. . . . . . 7 ⊢ ((Base‘𝐶) × (Base‘𝐷)) ∈ V
16		resfunexg 7073	. . . . . . 7 ⊢ ((Fun 1^st ∧ ((Base‘𝐶) × (Base‘𝐷)) ∈ V) → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V)
17	12, 15, 16	mp2an 688	. . . . . 6 ⊢ (1^st ↾ ((Base‘𝐶) × (Base‘𝐷))) ∈ V
18	15, 15	mpoex 7893	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦))) ∈ V
19	17, 18	op2ndd 7815	. . . . 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 4808	. . 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 2781	. 2 ⊢ (𝜑 → 𝑃 = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (2^nd ‘𝑃)⟩)
23		eqid 2738	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
24		eqid 2738	. . . 4 ⊢ (Id‘𝑇) = (Id‘𝑇)
25		eqid 2738	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
26		eqid 2738	. . . 4 ⊢ (comp‘𝑇) = (comp‘𝑇)
27		eqid 2738	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
28	1, 6, 7	xpccat 17823	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
29		f1stres 7828	. . . . 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 2738	. . . . . 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 7288	. . . . . . 7 ⊢ (𝑥(Hom ‘𝑇)𝑦) ∈ V
33		resfunexg 7073	. . . . . . 7 ⊢ ((Fun 1^st ∧ (𝑥(Hom ‘𝑇)𝑦) ∈ V) → (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V)
34	12, 32, 33	mp2an 688	. . . . . 6 ⊢ (1^st ↾ (𝑥(Hom ‘𝑇)𝑦)) ∈ V
35	31, 34	fnmpoi 7883	. . . . 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 6510	. . . . 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 257	. . . 4 ⊢ (𝜑 → (2^nd ‘𝑃) Fn (((Base‘𝐶) × (Base‘𝐷)) × ((Base‘𝐶) × (Base‘𝐷))))
38		f1stres 7828	. . . . . 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 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 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
42		simprr 769	. . . . . . . . 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 17826	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	1, 4, 23, 44, 5, 41, 42	xpchom 17813	. . . . . . . . 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 5880	. . . . . . . 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 2778	. . . . . . 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 6569	. . . . . 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 257	. . . . 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 6775	. . . . . . . 8 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
51	50	ad2antrl 724	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
52		fvres 6775	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑦) = (1^st ‘𝑦))
53	52	ad2antll 725	. . . . . . 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 7273	. . . . . 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 6579	. . . . 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 256	. . . 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 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 17308	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) ∈ (𝑥(Hom ‘𝑇)𝑥))
60	59	fvresd 6776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ (𝑥(Hom ‘𝑇)𝑥))‘((Id‘𝑇)‘𝑥)) = (1^st ‘((Id‘𝑇)‘𝑥)))
61		1st2nd2 7843	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
62	61	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
63	62	fveq2d 6760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ((Id‘𝑇)‘⟨(1^st ‘𝑥), (2^nd ‘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 2738	. . . . . . . . 9 ⊢ (Id‘𝐷) = (Id‘𝐷)
67		xp1st 7836	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑥) ∈ (Base‘𝐶))
68	67	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ‘𝑥) ∈ (Base‘𝐶))
69		xp2nd 7837	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
70	69	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (2^nd ‘𝑥) ∈ (Base‘𝐷))
71	1, 64, 65, 2, 3, 25, 66, 24, 68, 70	xpcid 17822	. . . . . . . 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 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((Id‘𝑇)‘𝑥) = ⟨((Id‘𝐶)‘(1^st ‘𝑥)), ((Id‘𝐷)‘(2^nd ‘𝑥))⟩)
73		fvex 6769	. . . . . . . 8 ⊢ ((Id‘𝐶)‘(1^st ‘𝑥)) ∈ V
74		fvex 6769	. . . . . . . 8 ⊢ ((Id‘𝐷)‘(2^nd ‘𝑥)) ∈ V
75	73, 74	op1std 7814	. . . . . . 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 2778	. . . . 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 17826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(2^nd ‘𝑃)𝑥) = (1^st ↾ (𝑥(Hom ‘𝑇)𝑥)))
79	78	fveq1d 6758	. . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → ((1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))‘𝑥) = (1^st ‘𝑥))
81	80	fveq2d 6760	. . . . 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 2788	. . . 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 1131	. . . . . . . 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 1204	. . . . . . . 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 1205	. . . . . . . 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 1206	. . . . . . . 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 1199	. . . . . . . 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 1200	. . . . . . . 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 17311	. . . . . . 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 6776	. . . . . 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 17817	. . . . . 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 2778	. . . . 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 1131	. . . . . . 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 1131	. . . . . . 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 17826	. . . . . 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 6758	. . . . 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 6776	. . . . . . . 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 6776	. . . . . . . 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 4809	. . . . . . 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 6776	. . . . . . 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 7273	. . . . . 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 17826	. . . . . . . 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 6758	. . . . . . 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 6776	. . . . . . 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 2778	. . . . . 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 17826	. . . . . . . 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 6758	. . . . . . 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 6776	. . . . . . 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 2778	. . . . . 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 7276	. . . . 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 2788	. . . 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 17496	. . 3 ⊢ (𝜑 → (1^st ↾ ((Base‘𝐶) × (Base‘𝐷)))(𝑇 Func 𝐶)(2^nd ‘𝑃))
113		df-br 5071	. . 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 217	. 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ⟨cop 4564 class class class wbr 5070 × cxp 5578 ↾ cres 5582 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 –onto→wfo 6416 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 Hom chom 16899 compcco 16900 Catccat 17290 Idccid 17291 Func cfunc 17485 ×_c cxpc 17801 1^st_F c1stf 17802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-hom 16912 df-cco 16913 df-cat 17294 df-cid 17295 df-func 17489 df-xpc 17805 df-1stf 17806

This theorem is referenced by: prf1st 17837 1st2ndprf 17839 uncfcl 17869 uncf1 17870 uncf2 17871 diagcl 17875 diag11 17877 diag12 17878 diag2 17879 yonedalem1 17906 yonedalem21 17907 yonedalem22 17912

W3C validator