termfucterm - Mathbox for Zhi Wang

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Theorem termfucterm 49555

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

**Hypotheses**
Ref	Expression
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**
Ref	Expression
termfucterm	⊢ (𝜑 → (𝑋 Func 𝑌) = (𝑋𝐼𝑌))

Proof of Theorem termfucterm Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		termfucterm.yt	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ TermCat)
2		termfucterm.c	. . . . . . . 8 ⊢ 𝐶 = (CatCat‘𝑈)
3		termfucterm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
4		termfucterm.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		termfucterm.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		termfucterm.xt	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ TermCat)
7	2, 3, 4, 5, 6	termcciso 49527	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ TermCat ↔ 𝑋( ≃_𝑐 ‘𝐶)𝑌))
8	1, 7	mpbid 232	. . . . . 6 ⊢ (𝜑 → 𝑋( ≃_𝑐 ‘𝐶)𝑌)
9		termfucterm.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
10		cicrcl2 49054	. . . . . . . 8 ⊢ (𝑋( ≃_𝑐 ‘𝐶)𝑌 → 𝐶 ∈ Cat)
11	8, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	9, 3, 11, 4, 5	cic 17698	. . . . . 6 ⊢ (𝜑 → (𝑋( ≃_𝑐 ‘𝐶)𝑌 ↔ ∃𝑔 𝑔 ∈ (𝑋𝐼𝑌)))
13	8, 12	mpbid 232	. . . . 5 ⊢ (𝜑 → ∃𝑔 𝑔 ∈ (𝑋𝐼𝑌))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) → ∃𝑔 𝑔 ∈ (𝑋𝐼𝑌))
15		eqid 2730	. . . . . . . 8 ⊢ (𝑋 FuncCat 𝑌) = (𝑋 FuncCat 𝑌)
16	6	termccd 49490	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Cat)
17	15, 16, 1	fucterm 49553	. . . . . . 7 ⊢ (𝜑 → (𝑋 FuncCat 𝑌) ∈ TermCat)
18	17	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → (𝑋 FuncCat 𝑌) ∈ TermCat)
19	15	fucbas 17862	. . . . . 6 ⊢ (𝑋 Func 𝑌) = (Base‘(𝑋 FuncCat 𝑌))
20		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Func 𝑌))
21		fullfunc 17807	. . . . . . 7 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
22		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑋) = (Base‘𝑋)
23		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑌) = (Base‘𝑌)
24		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋𝐼𝑌))
25	2, 22, 23, 9, 24	catcisoi 49411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → (𝑔 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝑔):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
26	25	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
27	26	elin1d 4152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Full 𝑌))
28	21, 27	sselid 3930	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Func 𝑌))
29	18, 19, 20, 28	termcbasmo 49494	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑓 = 𝑔)
30	29, 24	eqeltrd 2829	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋𝐼𝑌))
31	14, 30	exlimddv 1936	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) → 𝑓 ∈ (𝑋𝐼𝑌))
32		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋𝐼𝑌))
33	2, 22, 23, 9, 32	catcisoi 49411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → (𝑓 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝑓):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
34	33	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
35	34	elin1d 4152	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Full 𝑌))
36	21, 35	sselid 3930	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Func 𝑌))
37	31, 36	impbida 800	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋 Func 𝑌) ↔ 𝑓 ∈ (𝑋𝐼𝑌)))
38	37	eqrdv 2728	1 ⊢ (𝜑 → (𝑋 Func 𝑌) = (𝑋𝐼𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2110 ∩ cin 3899 class class class wbr 5089 –1-1-onto→wf1o 6476 ‘cfv 6477 (class class class)co 7341 1^st c1st 7914 Basecbs 17112 Catccat 17562 Isociso 17645 ≃_𝑐 ccic 17694 Func cfunc 17753 Full cful 17803 Faith cfth 17804 FuncCat cfuc 17844 CatCatccatc 17997 TermCatctermc 49483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 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 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-fz 13400 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-hom 17177 df-cco 17178 df-cat 17566 df-cid 17567 df-homf 17568 df-comf 17569 df-oppc 17610 df-sect 17646 df-inv 17647 df-iso 17648 df-cic 17695 df-func 17757 df-idfu 17758 df-cofu 17759 df-full 17805 df-fth 17806 df-nat 17845 df-fuc 17846 df-inito 17883 df-termo 17884 df-catc 17998 df-thinc 49429 df-termc 49484

This theorem is referenced by: (None)

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