termfucterm

Description: All functors between two terminal categories are isomorphisms. (Contributed by Zhi Wang, 17-Nov-2025)

	Ref	Expression
Hypotheses	termfucterm.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
	termfucterm.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	termfucterm.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
	termfucterm.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	termfucterm.xt	⊢ ( 𝜑 → 𝑋 ∈ TermCat )
	termfucterm.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	termfucterm.yt	⊢ ( 𝜑 → 𝑌 ∈ TermCat )
Assertion	termfucterm	⊢ ( 𝜑 → ( 𝑋 Func 𝑌 ) = ( 𝑋 𝐼 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		termfucterm.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2		termfucterm.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		termfucterm.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
4		termfucterm.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		termfucterm.xt	⊢ ( 𝜑 → 𝑋 ∈ TermCat )
6		termfucterm.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		termfucterm.yt	⊢ ( 𝜑 → 𝑌 ∈ TermCat )
8	1 2 4 6 5	termcciso	⊢ ( 𝜑 → ( 𝑌 ∈ TermCat ↔ 𝑋 ( ≃_𝑐 ‘ 𝐶 ) 𝑌 ) )
9	7 8	mpbid	⊢ ( 𝜑 → 𝑋 ( ≃_𝑐 ‘ 𝐶 ) 𝑌 )
10		cicrcl2	⊢ ( 𝑋 ( ≃_𝑐 ‘ 𝐶 ) 𝑌 → 𝐶 ∈ Cat )
11	9 10	syl	⊢ ( 𝜑 → 𝐶 ∈ Cat )
12	3 2 11 4 6	cic	⊢ ( 𝜑 → ( 𝑋 ( ≃_𝑐 ‘ 𝐶 ) 𝑌 ↔ ∃ 𝑔 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) )
13	9 12	mpbid	⊢ ( 𝜑 → ∃ 𝑔 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) )
14	13	adantr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) → ∃ 𝑔 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) )
15		eqid	⊢ ( 𝑋 FuncCat 𝑌 ) = ( 𝑋 FuncCat 𝑌 )
16	5	termccatd	⊢ ( 𝜑 → 𝑋 ∈ Cat )
17	15 16 7	fucterm	⊢ ( 𝜑 → ( 𝑋 FuncCat 𝑌 ) ∈ TermCat )
18	17	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( 𝑋 FuncCat 𝑌 ) ∈ TermCat )
19	15	fucbas	⊢ ( 𝑋 Func 𝑌 ) = ( Base ‘ ( 𝑋 FuncCat 𝑌 ) )
20		simplr	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( 𝑋 Func 𝑌 ) )
21		fullfunc	⊢ ( 𝑋 Full 𝑌 ) ⊆ ( 𝑋 Func 𝑌 )
22		eqid	⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
23		eqid	⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 )
24		simpr	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) )
25	1 22 23 3 24	catcisoi	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( 𝑔 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) ∧ ( 1^st ‘ 𝑔 ) : ( Base ‘ 𝑋 ) –1-1-onto→ ( Base ‘ 𝑌 ) ) )
26	25	simpld	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑔 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) )
27	26	elin1d	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑔 ∈ ( 𝑋 Full 𝑌 ) )
28	21 27	sselid	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑔 ∈ ( 𝑋 Func 𝑌 ) )
29	18 19 20 28	termcbasmo	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 = 𝑔 )
30	29 24	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) ∧ 𝑔 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) )
31	14 30	exlimddv	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 Func 𝑌 ) ) → 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) )
32		simpr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) )
33	1 22 23 3 32	catcisoi	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( 𝑓 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) ∧ ( 1^st ‘ 𝑓 ) : ( Base ‘ 𝑋 ) –1-1-onto→ ( Base ‘ 𝑌 ) ) )
34	33	simpld	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) )
35	34	elin1d	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( 𝑋 Full 𝑌 ) )
36	21 35	sselid	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑓 ∈ ( 𝑋 Func 𝑌 ) )
37	31 36	impbida	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 Func 𝑌 ) ↔ 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) )
38	37	eqrdv	⊢ ( 𝜑 → ( 𝑋 Func 𝑌 ) = ( 𝑋 𝐼 𝑌 ) )

Metamath Proof Explorer

Theorem termfucterm

Proof