Theorem lmdran 50161

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 50024). (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 50145	. . 3 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
2	1	breqi 5092	. 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 49675	. . . . . 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 2737	. . . . . 6 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
6		eqid 2737	. . . . . . 7 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
7	6	fucbas 17924	. . . . . 6 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
8	4, 5, 7	oppcuprcl3 49690	. . . . 5 ⊢ ((𝜑 ∧ 𝑋(( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝐹 ∈ (𝐷 Func 𝐶))
9		eqid 2737	. . . . . 6 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
10		eqid 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	4, 9, 10	oppcuprcl4 49689	. . . . 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 49675	. . . . . 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 49690	. . . . 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 2737	. . . . . . . . . 10 ⊢ (oppCat‘( 1 FuncCat 𝐶)) = (oppCat‘( 1 FuncCat 𝐶))
19		eqid 2737	. . . . . . . . . . 11 ⊢ ( 1 FuncCat 𝐶) = ( 1 FuncCat 𝐶)
20	19	fucbas 17924	. . . . . . . . . 10 ⊢ ( 1 Func 𝐶) = (Base‘( 1 FuncCat 𝐶))
21	14, 18, 20	oppcuprcl4 49689	. . . . . . . . 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 17823	. . . . . . . . 9 ⊢ Rel ( 1 Func 𝐶)
23	21, 22	oppfrcllem 49617	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → 𝑌 ≠ ∅)
24	17, 23	eqnetrrd 3001	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌(( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺))((oppCat‘( 1 FuncCat 𝐶)) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑀) → ((1st ‘𝐿)‘𝑋) ≠ ∅)
25		fvfundmfvn0 6875	. . . . . . . 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 49566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
32	31	funcrcl3 49570	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → 𝐶 ∈ Cat)
33		lmdran.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc 1 )
34	10, 29, 32, 33	diag1f1o 50024	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
35		f1of 6775	. . . . . . . . 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 6673	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐷 Func 𝐶)) → dom (1st ‘𝐿) = (Base‘𝐶))
38	15, 37	syldan 592	. . . . . 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 2839	. . . . 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 17678	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
42	18, 20	oppcbas 17678	. . . . 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 718	. . . . . . . . 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 49969	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 1 ∈ Cat)
47	33, 44, 46, 19	diagcl 18201	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝐿 ∈ (𝐶 Func ( 1 FuncCat 𝐶)))
48	47	oppf1 49629	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
49	48	fveq1d 6837	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
50	43, 49	eqtr4d 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑌 = ((1st ‘( oppFunc ‘𝐿))‘𝑋))
51		eqid 2737	. . . . . . 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 2738	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺))
55	33, 51, 53, 44, 54	prcofdiag 49884	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → ((⟨ 1 , 𝐶⟩ −∘F 𝐺) ∘func 𝐿) = (𝐶Δfunc𝐷))
56	19, 44, 6, 53	prcoffunca 49876	. . . . . 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 49640	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (( oppFunc ‘(⟨ 1 , 𝐶⟩ −∘F 𝐺)) ∘func ( oppFunc ‘𝐿)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
58		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → 𝑋 ∈ (Base‘𝐶))
59	18, 5, 56	oppfoppc2 49632	. . . . 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 50028	. . . . . 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 49655	. . . . 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 718	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑋 ∈ (Base‘𝐶))) → (1st ‘𝐿):(Base‘𝐶)–1-1-onto→( 1 Func 𝐶))
63	48	f1oeq1d 6770	. . . . . . 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 6782	. . . . . 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 49712	. . . 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 840	. . 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 2737	. . . . . 6 ⊢ (⟨ 1 , 𝐶⟩ −∘F 𝐺) = (⟨ 1 , 𝐶⟩ −∘F 𝐺)
70	18, 5, 69	ranval3 50121	. . . . 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 5097	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  {csn 4568  ⟨cop 4574   class class class wbr 5086  dom cdm 5625   ↾ cres 5627  Fun wfun 6487  ⟶wf 6489  –onto→wfo 6491  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  Basecbs 17173  Catccat 17624  oppCatcoppc 17671   Func cfunc 17815   FuncCat cfuc 17906  Δ_funccdiag 18172   oppFunc coppf 49612   UP cup 49663   −∘_F cprcof 49863  TermCatctermc 49962   Ran cran 50096   Limit clmd 50133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-hom 17238  df-cco 17239  df-cat 17628  df-cid 17629  df-oppc 17672  df-func 17819  df-cofu 17821  df-full 17867  df-fth 17868  df-nat 17907  df-fuc 17908  df-xpc 18132  df-1stf 18133  df-curf 18174  df-diag 18176  df-oppf 49613  df-up 49664  df-swapf 49750  df-fuco 49807  df-prcof 49864  df-thinc 49908  df-termc 49963  df-ran 50098  df-lmd 50135

This theorem is referenced by: (None)