Theorem lmdran 50469

Description: To each limit of a diagram there is a corresponding right Kan extention of the diagram along a functor to a terminal category. The morphism parts coincide, while the object parts are one-to-one correspondent (diag1f1o 50332). (Contributed by Zhi Wang, 26-Nov-2025.)

Hypotheses

Ref	Expression
lmdran.1	⊢ (𝜑 → 1 ∈ TermCat)
lmdran.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
lmdran.l	⊢ 𝐿 = (𝐶Δfunc 1 )
lmdran.y	⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))

Assertion

Ref	Expression
lmdran	⊢ (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))

Proof of Theorem lmdran

Step	Hyp	Ref	Expression
1		lmdfval2 50453	. . 3 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
2	1	breqi 5115	. 2 ⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
3		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
4	3	up1st2nd 49983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋(⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
5		eqid 2763	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
6		eqid 2763	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
7	6	fucbas 18015	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
8	4, 5, 7	oppcuprcl3 49998	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
9		eqid 2763	. . . . . 6 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
10		eqid 2763	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	4, 9, 10	oppcuprcl4 49997	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
12	8, 11	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
13		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
14	13	up1st2nd 49983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌(⟨(1st ‘( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))), (2nd ‘( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)))⟩((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
15	14, 5, 7	oppcuprcl3 49998	. . . . 5 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
16		lmdran.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 = ((1st ‘𝐿)‘𝑋))
17	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 = ((1st ‘𝐿)‘𝑋))
18		eqid 2763	. . . . . . . . . 10 ⊢ (oppCat‘( 1 FuncCat 𝐶)) = (oppCat‘( 1 FuncCat 𝐶))
19		eqid 2763	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
20	19	fucbas 18015	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
21	14, 18, 20	oppcuprcl4 49997	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 ∈ ( 1 Func 𝐶))
22		relfunc 17914	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
23	21, 22	oppfrcllem 49925	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 ≠ ∅)
24	17, 23	eqnetrrd 3026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
25		fvfundmfvn0 6921	. . . . . . . 8 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → (𝑋 ∈ dom (1st ‘𝐿) ∧ Fun ((1st ‘𝐿) ↾ {𝑋})))
26	25	simpld 499	. . . . . . 7 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → 𝑋 ∈ dom (1st ‘𝐿))
27	24, 26	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋 ∈ dom (1st ‘𝐿))
28		lmdran.1	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ TermCat)
29	28	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 1 ∈ TermCat)
30		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
31	30	func1st2nd 49874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
32	31	funcrcl3 49878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
33		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
34	10, 29, 32, 33	diag1f1o 50332	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
35		f1of 6820	. . . . . . . . 9 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
36	34, 35	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
37	36	fdmd 6716	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
38	15, 37	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → dom (1st ‘𝐿) = (Base‘𝐶))
39	27, 38	eleqtrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
40	15, 39	jca 520	. . . 4 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
41	9, 10	oppcbas 17769	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
42	18, 20	oppcbas 17769	. . . . 5 ⊢ ( 1 Func 𝐶) = (Base‘(oppCat‘( 1 FuncCat 𝐶)))
43	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘𝐿)‘𝑋))
44	32	adantrr 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
45	28	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ TermCat)
46	45	termccd 50277	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ Cat)
47	33, 44, 46, 19	diagcl 18292	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ (𝐶 Func ( 1 FuncCat 𝐶)))
48	47	oppf1 49937	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
49	48	fveq1d 6883	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
50	43, 49	eqtr4d 2801	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘( oppFunc ‘𝐿))‘𝑋))
51		eqid 2763	. . . . . . 7 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
52		lmdran.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
53	52	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 1 ))
54		eqidd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
55	33, 51, 53, 44, 54	prcofdiag 50192	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
56	19, 44, 6, 53	prcoffunca 50184	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) ∈ (( 1 FuncCat 𝐶) Func (𝐷 FuncCat 𝐶)))
57	55, 47, 56	cofuoppf 49948	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)) ∘func ( oppFunc ‘𝐿)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
58		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
59	18, 5, 56	oppfoppc2 49940	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)) ∈ ((oppCat‘( 1 FuncCat 𝐶)) Func (oppCat‘(𝐷 FuncCat 𝐶))))
60	44, 45, 19, 33	diagffth 50336	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ ((𝐶 Full ( 1 FuncCat 𝐶)) ∩ (𝐶 Faith ( 1 FuncCat 𝐶))))
61	9, 18, 60	ffthoppf 49963	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ( oppFunc ‘𝐿) ∈ (((oppCat‘𝐶) Full (oppCat‘( 1 FuncCat 𝐶))) ∩ ((oppCat‘𝐶) Faith (oppCat‘( 1 FuncCat 𝐶)))))
62	34	adantrr 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
63	48	f1oeq1d 6815	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) ↔ (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶)))
64	62, 63	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
65		f1ofo 6828	. . . . . 6 ⊢ ((1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–onto→( 1 Func 𝐶))
66	64, 65	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–onto→( 1 Func 𝐶))
67	41, 42, 50, 57, 58, 59, 61, 66	uptr2a 50020	. . . 4 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀 ↔ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀))
68	12, 40, 67	bibiad 852	. . 3 ⊢ (𝜑 → (𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀 ↔ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀))
69		eqid 2763	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
70	18, 5, 69	ranval3 50429	. . . . 5 ⊢ (𝐺 ∈ (𝐷 Func 1 ) → (𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹) = (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹))
71	52, 70	syl 18	. . . 4 ⊢ (𝜑 → (𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹) = (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹))
72	71	breqd 5120	. . 3 ⊢ (𝜑 → (𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀 ↔ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀))
73	68, 72	bitr4d 285	. 2 ⊢ (𝜑 → (𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))
74	2, 73	bitrid 286	1 ⊢ (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∅c0 4286  {csn 4589  ⟨cop 4595   class class class wbr 5109  dom cdm 5661   ↾ cres 5663  Fun wfun 6530  ⟶wf 6532  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17264  Catccat 17715  oppCatcoppc 17762   Func cfunc 17906   FuncCat cfuc 17997  Δ_funccdiag 18263   oppFunc coppf 49920   UP cup 49971   −∘_F cprcof 50171  TermCatctermc 50270   Ran cran 50404   Limit clmd 50441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-oppc 17763  df-func 17910  df-cofu 17912  df-full 17958  df-fth 17959  df-nat 17998  df-fuc 17999  df-xpc 18223  df-1stf 18224  df-curf 18265  df-diag 18267  df-oppf 49921  df-up 49972  df-swapf 50058  df-fuco 50115  df-prcof 50172  df-thinc 50216  df-termc 50271  df-ran 50406  df-lmd 50443

This theorem is referenced by: (None)