Theorem lmdran 50292

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 50155). (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 50276	. . 3 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
2	1	breqi 5106	. 2 ⊢ (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀)
3		simpr 488	. . . . . . 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 49806	. . . . . 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 2762	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
6		eqid 2762	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
7	6	fucbas 17996	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
8	4, 5, 7	oppcuprcl3 49821	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
9		eqid 2762	. . . . . 6 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
10		eqid 2762	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	4, 9, 10	oppcuprcl4 49820	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
12	8, 11	jca 519	. . . 4 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
13		simpr 488	. . . . . . 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 49806	. . . . . 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 49821	. . . . 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 484	. . . . . . . 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 2762	. . . . . . . . . 10 ⊢ (oppCat‘( 1 FuncCat 𝐶)) = (oppCat‘( 1 FuncCat 𝐶))
19		eqid 2762	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
20	19	fucbas 17996	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
21	14, 18, 20	oppcuprcl4 49820	. . . . . . . . 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 17895	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
23	21, 22	oppfrcllem 49748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 ≠ ∅)
24	17, 23	eqnetrrd 3025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
25		fvfundmfvn0 6907	. . . . . . . 8 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → (𝑋 ∈ dom (1st ‘𝐿) ∧ Fun ((1st ‘𝐿) ↾ {𝑋})))
26	25	simpld 498	. . . . . . 7 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → 𝑋 ∈ dom (1st ‘𝐿))
27	24, 26	syl 17	. . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 1 ∈ TermCat)
30		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
31	30	func1st2nd 49697	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
32	31	funcrcl3 49701	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
33		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
34	10, 29, 32, 33	diag1f1o 50155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
35		f1of 6806	. . . . . . . . 9 ⊢ ((1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)⟶( 1 Func 𝐶))
37	36	fdmd 6702	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
38	15, 37	syldan 600	. . . . . 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 2864	. . . . 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 519	. . . 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 17750	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
42	18, 20	oppcbas 17750	. . . . 5 ⊢ ( 1 Func 𝐶) = (Base‘(oppCat‘( 1 FuncCat 𝐶)))
43	16	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘𝐿)‘𝑋))
44	32	adantrr 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
45	28	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ TermCat)
46	45	termccd 50100	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ Cat)
47	33, 44, 46, 19	diagcl 18273	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ (𝐶 Func ( 1 FuncCat 𝐶)))
48	47	oppf1 49760	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
49	48	fveq1d 6869	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
50	43, 49	eqtr4d 2800	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘( oppFunc ‘𝐿))‘𝑋))
51		eqid 2762	. . . . . . 7 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
52		lmdran.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
53	52	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 1 ))
54		eqidd 2763	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
55	33, 51, 53, 44, 54	prcofdiag 50015	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
56	19, 44, 6, 53	prcoffunca 50007	. . . . . 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 49771	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)) ∘func ( oppFunc ‘𝐿)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
58		simprr 782	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
59	18, 5, 56	oppfoppc2 49763	. . . . 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 50159	. . . . . 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 49786	. . . . 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 727	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
63	48	f1oeq1d 6801	. . . . . . 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 259	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
65		f1ofo 6814	. . . . . 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 17	. . . . 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 49843	. . . 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 850	. . 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 2762	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
70	18, 5, 69	ranval3 50252	. . . . 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 17	. . . 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 5111	. . 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 284	. 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 285	1 ⊢ (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∅c0 4285  {csn 4582  ⟨cop 4588   class class class wbr 5100  dom cdm 5647   ↾ cres 5649  Fun wfun 6515  ⟶wf 6517  –onto→wfo 6519  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17245  Catccat 17696  oppCatcoppc 17743   Func cfunc 17887   FuncCat cfuc 17978  Δ_funccdiag 18244   oppFunc coppf 49743   UP cup 49794   −∘_F cprcof 49994  TermCatctermc 50093   Ran cran 50227   Limit clmd 50264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-er 8678  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-hom 17310  df-cco 17311  df-cat 17700  df-cid 17701  df-oppc 17744  df-func 17891  df-cofu 17893  df-full 17939  df-fth 17940  df-nat 17979  df-fuc 17980  df-xpc 18204  df-1stf 18205  df-curf 18246  df-diag 18248  df-oppf 49744  df-up 49795  df-swapf 49881  df-fuco 49938  df-prcof 49995  df-thinc 50039  df-termc 50094  df-ran 50229  df-lmd 50266

This theorem is referenced by: (None)