Theorem lmdran 49650

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 49513). (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 49634	. . 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 484	. . . . . . 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 49164	. . . . . 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 2730	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
6		eqid 2730	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
7	6	fucbas 17931	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
8	4, 5, 7	oppcuprcl3 49179	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
9		eqid 2730	. . . . . 6 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
10		eqid 2730	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	4, 9, 10	oppcuprcl4 49178	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑋 ∈ (Base‘𝐶))
12	8, 11	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶)))
13		simpr 484	. . . . . . 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 49164	. . . . . 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 49179	. . . . 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 480	. . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (oppCat‘( 1 FuncCat 𝐶)) = (oppCat‘( 1 FuncCat 𝐶))
19		eqid 2730	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
20	19	fucbas 17931	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
21	14, 18, 20	oppcuprcl4 49178	. . . . . . . . 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 17830	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
23	21, 22	oppfrcllem 49106	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 ≠ ∅)
24	17, 23	eqnetrrd 2994	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
25		fvfundmfvn0 6903	. . . . . . . 8 ⊢ (((1st ‘𝐿)‘𝑋) ≠ ∅ → (𝑋 ∈ dom (1st ‘𝐿) ∧ Fun ((1st ‘𝐿) ↾ {𝑋})))
26	25	simpld 494	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 1 ∈ TermCat)
30		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐹 ∈ (𝐷 Func 𝐶))
31	30	func1st2nd 49055	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
32	31	funcrcl3 49059	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
33		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
34	10, 29, 32, 33	diag1f1o 49513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
35		f1of 6802	. . . . . . . . 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 6700	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
38	15, 37	syldan 591	. . . . . 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 2831	. . . . 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 511	. . . 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 17685	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
42	18, 20	oppcbas 17685	. . . . 5 ⊢ ( 1 Func 𝐶) = (Base‘(oppCat‘( 1 FuncCat 𝐶)))
43	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘𝐿)‘𝑋))
44	32	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
45	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ TermCat)
46	45	termccd 49458	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ Cat)
47	33, 44, 46, 19	diagcl 18208	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ (𝐶 Func ( 1 FuncCat 𝐶)))
48	47	oppf1 49118	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
49	48	fveq1d 6862	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
50	43, 49	eqtr4d 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘( oppFunc ‘𝐿))‘𝑋))
51		eqid 2730	. . . . . . 7 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
52		lmdran.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 1 ))
53	52	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐷 Func 1 ))
54		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
55	33, 51, 53, 44, 54	prcofdiag 49373	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
56	19, 44, 6, 53	prcoffunca 49365	. . . . . 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 49129	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)) ∘func ( oppFunc ‘𝐿)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
58		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
59	18, 5, 56	oppfoppc2 49121	. . . . 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 49517	. . . . . 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 49144	. . . . 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 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
63	48	f1oeq1d 6797	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
65		f1ofo 6809	. . . . . 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 49201	. . . 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 839	. . 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 2730	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
70	18, 5, 69	ranval3 49610	. . . . 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 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 282	. 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 283	1 ⊢ (𝜑 → (𝑋((𝐶 Limit 𝐷)‘𝐹)𝑀 ↔ 𝑌(𝐺(⟨𝐷, 1 ⟩ Ran 𝐶)𝐹)𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4298  {csn 4591  ⟨cop 4597   class class class wbr 5109  dom cdm 5640   ↾ cres 5642  Fun wfun 6507  ⟶wf 6509  –onto→wfo 6511  –1-1-onto→wf1o 6512  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17185  Catccat 17631  oppCatcoppc 17678   Func cfunc 17822   FuncCat cfuc 17913  Δ_funccdiag 18179   oppFunc coppf 49101   UP cup 49152   −∘_F cprcof 49352  TermCatctermc 49451   Ran cran 49585   Limit clmd 49622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-oppc 17679  df-func 17826  df-cofu 17828  df-full 17874  df-fth 17875  df-nat 17914  df-fuc 17915  df-xpc 18139  df-1stf 18140  df-curf 18181  df-diag 18183  df-oppf 49102  df-up 49153  df-swapf 49239  df-fuco 49296  df-prcof 49353  df-thinc 49397  df-termc 49452  df-ran 49587  df-lmd 49624

This theorem is referenced by: (None)