termfucterm - Mathbox for Zhi Wang

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Theorem termfucterm 49533

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 49505	. . . . . . 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 49032	. . . . . . . 8 ⊢ (𝑋( ≃_𝑐 ‘𝐶)𝑌 → 𝐶 ∈ Cat)
11	8, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	9, 3, 11, 4, 5	cic 17761	. . . . . 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 2729	. . . . . . . 8 ⊢ (𝑋 FuncCat 𝑌) = (𝑋 FuncCat 𝑌)
16	6	termccd 49468	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Cat)
17	15, 16, 1	fucterm 49531	. . . . . . 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 17925	. . . . . 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 17870	. . . . . . 7 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
22		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑋) = (Base‘𝑋)
23		eqid 2729	. . . . . . . . . 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 49389	. . . . . . . . 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 4167	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Full 𝑌))
28	21, 27	sselid 3944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑔 ∈ (𝑋 Func 𝑌))
29	18, 19, 20, 28	termcbasmo 49472	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝑋 Func 𝑌)) ∧ 𝑔 ∈ (𝑋𝐼𝑌)) → 𝑓 = 𝑔)
30	29, 24	eqeltrd 2828	. . . 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 49389	. . . . . 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 4167	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Full 𝑌))
36	21, 35	sselid 3944	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐼𝑌)) → 𝑓 ∈ (𝑋 Func 𝑌))
37	31, 36	impbida 800	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋 Func 𝑌) ↔ 𝑓 ∈ (𝑋𝐼𝑌)))
38	37	eqrdv 2727	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 3913 class class class wbr 5107 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 Basecbs 17179 Catccat 17625 Isociso 17708 ≃_𝑐 ccic 17757 Func cfunc 17816 Full cful 17866 Faith cfth 17867 FuncCat cfuc 17907 CatCatccatc 18060 TermCatctermc 49461

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-cid 17630 df-homf 17631 df-comf 17632 df-oppc 17673 df-sect 17709 df-inv 17710 df-iso 17711 df-cic 17758 df-func 17820 df-idfu 17821 df-cofu 17822 df-full 17868 df-fth 17869 df-nat 17908 df-fuc 17909 df-inito 17946 df-termo 17947 df-catc 18061 df-thinc 49407 df-termc 49462

This theorem is referenced by: (None)

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