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	$⊢ φ \to X \in B$
	termfucterm.xt	No typesetting found for \|- ( ph -> X e. TermCat ) with typecode \|-
	termfucterm.y	$⊢ φ \to Y \in B$
	termfucterm.yt	No typesetting found for \|- ( ph -> Y e. TermCat ) with typecode \|-
Assertion	termfucterm	$⊢ φ \to X Func Y = X I Y$

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	$⊢ φ \to X \in B$
5		termfucterm.xt	Could not format ( ph -> X e. TermCat ) : No typesetting found for \|- ( ph -> X e. TermCat ) with typecode \|-
6		termfucterm.y	$⊢ φ \to Y \in B$
7		termfucterm.yt	Could not format ( ph -> Y e. TermCat ) : No typesetting found for \|- ( ph -> Y e. TermCat ) with typecode \|-
8	1 2 4 6 5	termcciso	Could not format ( ph -> ( Y e. TermCat <-> X ( ~=c ` C ) Y ) ) : No typesetting found for \|- ( ph -> ( Y e. TermCat <-> X ( ~=c ` C ) Y ) ) with typecode \|-
9	7 8	mpbid	$⊢ φ \to X ≃_{𝑐} (C) Y$
10		cicrcl2	$⊢ X ≃_{𝑐} (C) Y \to C \in Cat$
11	9 10	syl	$⊢ φ \to C \in Cat$
12	3 2 11 4 6	cic	$⊢ φ \to (X ≃_{𝑐} (C) Y \leftrightarrow \exists g g \in (X I Y))$
13	9 12	mpbid	$⊢ φ \to \exists g g \in (X I Y)$
14	13	adantr	$⊢ (φ \land f \in (X Func Y)) \to \exists g g \in (X I Y)$
15		eqid	$⊢ X FuncCat Y = X FuncCat Y$
16	5	termccd	$⊢ φ \to X \in Cat$
17	15 16 7	fucterm	Could not format ( ph -> ( X FuncCat Y ) e. TermCat ) : No typesetting found for \|- ( ph -> ( X FuncCat Y ) e. TermCat ) with typecode \|-
18	17	ad2antrr	Could not format ( ( ( ph /\ f e. ( X Func Y ) ) /\ g e. ( X I Y ) ) -> ( X FuncCat Y ) e. TermCat ) : No typesetting found for \|- ( ( ( ph /\ f e. ( X Func Y ) ) /\ g e. ( X I Y ) ) -> ( X FuncCat Y ) e. TermCat ) with typecode \|-
19	15	fucbas	$⊢ X Func Y = {Base}_{(X FuncCat Y)}$
20		simplr	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to f \in (X Func Y)$
21		fullfunc	$⊢ X Full Y \subseteq X Func Y$
22		eqid	$⊢ {Base}_{X} = {Base}_{X}$
23		eqid	$⊢ {Base}_{Y} = {Base}_{Y}$
24		simpr	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to g \in (X I Y)$
25	1 22 23 3 24	catcisoi	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to (g \in ((X Full Y) \cap (X Faith Y)) \land 1^{st} (g) : {Base}_{X} \underset{1-1 onto}{⟶} {Base}_{Y})$
26	25	simpld	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to g \in ((X Full Y) \cap (X Faith Y))$
27	26	elin1d	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to g \in (X Full Y)$
28	21 27	sselid	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to g \in (X Func Y)$
29	18 19 20 28	termcbasmo	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to f = g$
30	29 24	eqeltrd	$⊢ ((φ \land f \in (X Func Y)) \land g \in (X I Y)) \to f \in (X I Y)$
31	14 30	exlimddv	$⊢ (φ \land f \in (X Func Y)) \to f \in (X I Y)$
32		simpr	$⊢ (φ \land f \in (X I Y)) \to f \in (X I Y)$
33	1 22 23 3 32	catcisoi	$⊢ (φ \land f \in (X I Y)) \to (f \in ((X Full Y) \cap (X Faith Y)) \land 1^{st} (f) : {Base}_{X} \underset{1-1 onto}{⟶} {Base}_{Y})$
34	33	simpld	$⊢ (φ \land f \in (X I Y)) \to f \in ((X Full Y) \cap (X Faith Y))$
35	34	elin1d	$⊢ (φ \land f \in (X I Y)) \to f \in (X Full Y)$
36	21 35	sselid	$⊢ (φ \land f \in (X I Y)) \to f \in (X Func Y)$
37	31 36	impbida	$⊢ φ \to (f \in (X Func Y) \leftrightarrow f \in (X I Y))$
38	37	eqrdv	$⊢ φ \to X Func Y = X I Y$

Metamath Proof Explorer

Theorem termfucterm

Proof