termfucterm - Mathbox for Zhi Wang

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Theorem termfucterm 49422

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 49394	. . . . . . 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 48960	. . . . . . . 8 ⊢ (𝑋( ≃_𝑐 ‘𝐶)𝑌 → 𝐶 ∈ Cat)
11	8, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	9, 3, 11, 4, 5	cic 17767	. . . . . 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 49357	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Cat)
17	15, 16, 1	fucterm 49420	. . . . . . 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 17931	. . . . . 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 17876	. . . . . . 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 49292	. . . . . . . . 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 4175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Full 𝑌))
28	21, 27	sselid 3952	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Func 𝑌))
29	18, 19, 20, 28	termcbasmo 49361	. . . . 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 1935	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) → 𝑓 ∈ (𝑋𝐼𝑌))
32		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋𝐼𝑌))
33	2, 22, 23, 9, 32	catcisoi 49292	. . . . . 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 4175	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Full 𝑌))
36	21, 35	sselid 3952	. . 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 1540 ∃wex 1779 ∈ wcel 2109 ∩ cin 3921 class class class wbr 5115 –1-1-onto→wf1o 6518 ‘cfv 6519 (class class class)co 7394 1^st c1st 7975 Basecbs 17185 Catccat 17631 Isociso 17714 ≃_𝑐 ccic 17763 Func cfunc 17822 Full cful 17872 Faith cfth 17873 FuncCat cfuc 17913 CatCatccatc 18066 TermCatctermc 49350

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 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

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 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7851 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-er 8682 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-z 12546 df-dec 12666 df-uz 12810 df-fz 13482 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-homf 17637 df-comf 17638 df-oppc 17679 df-sect 17715 df-inv 17716 df-iso 17717 df-cic 17764 df-func 17826 df-idfu 17827 df-cofu 17828 df-full 17874 df-fth 17875 df-nat 17914 df-fuc 17915 df-inito 17952 df-termo 17953 df-catc 18067 df-thinc 49296 df-termc 49351

This theorem is referenced by: (None)

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