Theorem thincciso2 49945

Description: Categories isomorphic to a thin category are thin. Example 3.26(2) of [Adamek] p. 33. Note that "thincciso2.u" is redundant thanks to elbasfv 17176. (Contributed by Zhi Wang, 18-Oct-2025.)

Hypotheses

Ref	Expression
thincciso2.c	⊢ 𝐶 = (CatCat‘𝑈)
thincciso2.b	⊢ 𝐵 = (Base‘𝐶)
thincciso2.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
thincciso2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincciso2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincciso2.i	⊢ 𝐼 = (Iso‘𝐶)
thincciso2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
thincciso2.yt	⊢ (𝜑 → 𝑌 ∈ ThinCat)

Assertion

Ref	Expression
thincciso2	⊢ (𝜑 → 𝑋 ∈ ThinCat)

Proof of Theorem thincciso2

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2740	. 2 ⊢ (𝜑 → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2740	. 2 ⊢ (𝜑 → (Hom ‘𝑋) = (Hom ‘𝑋))
3		relfull 17868	. . . . . . . . . . . 12 ⊢ Rel (𝑋 Full 𝑌)
4		relin1 5755	. . . . . . . . . . . 12 ⊢ (Rel (𝑋 Full 𝑌) → Rel ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ Rel ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))
6		thincciso2.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
7		thincciso2.c	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (CatCat‘𝑈)
8		thincciso2.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐶)
9		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑋) = (Base‘𝑋)
10		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑌) = (Base‘𝑌)
11		thincciso2.u	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
12		thincciso2.x	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13		thincciso2.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		thincciso2.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (Iso‘𝐶)
15	7, 8, 9, 10, 11, 12, 13, 14	catciso 18069	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
16	6, 15	mpbid 233	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
17	16	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
18		1st2ndbr 7984	. . . . . . . . . . 11 ⊢ ((Rel ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))) → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
19	5, 17, 18	sylancr 593	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
20		eqid 2739	. . . . . . . . . . 11 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
21		eqid 2739	. . . . . . . . . . 11 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
22	9, 20, 21	isffth2 17876	. . . . . . . . . 10 ⊢ ((1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹) ↔ ((1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹) ∧ ∀𝑥 ∈ (Base‘𝑋)∀𝑦 ∈ (Base‘𝑋)(𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))))
23	19, 22	sylib 219	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹) ∧ ∀𝑥 ∈ (Base‘𝑋)∀𝑦 ∈ (Base‘𝑋)(𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦))))
24	23	simprd 496	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑋)∀𝑦 ∈ (Base‘𝑋)(𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
25	24	r19.21bi 3231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ∀𝑦 ∈ (Base‘𝑋)(𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
26	25	r19.21bi 3231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
27	26	anasss 467	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
28		ovex 7389	. . . . . 6 ⊢ (𝑥(Hom ‘𝑋)𝑦) ∈ V
29	28	f1oen 8909	. . . . 5 ⊢ ((𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝑋)𝑦)–1-1-onto→(((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) → (𝑥(Hom ‘𝑋)𝑦) ≈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
30	27, 29	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(Hom ‘𝑋)𝑦) ≈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
31		thincciso2.yt	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ThinCat)
32	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → 𝑌 ∈ ThinCat)
33	23	simpld 495	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
34	9, 10, 33	funcf1 17824	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝑋)⟶(Base‘𝑌))
35	34	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
36	35	adantrr 723	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
37	34	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
38	37	adantrl 722	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
39	32, 36, 38, 10, 21	thincmo 49918	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
40		modom2 9152	. . . . 5 ⊢ (∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ↔ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
41	39, 40	sylib 219	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
42		endomtr 8949	. . . 4 ⊢ (((𝑥(Hom ‘𝑋)𝑦) ≈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ∧ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o) → (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
43	30, 41, 42	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
44		modom2 9152	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦) ↔ (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
45	43, 44	sylibr 235	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦))
46	33	funcrcl2 49569	. 2 ⊢ (𝜑 → 𝑋 ∈ Cat)
47	1, 2, 45, 46	isthincd 49926	1 ⊢ (𝜑 → 𝑋 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3053   ∩ cin 3882   class class class wbr 5072  Rel wrel 5623  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  1_oc1o 8388   ≈ cen 8880   ≼ cdom 8881  Basecbs 17170  Hom chom 17222  Isociso 17704   Func cfunc 17812   Full cful 17862   Faith cfth 17863  CatCatccatc 18056  ThinCatcthinc 49907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-sect 17705  df-inv 17706  df-iso 17707  df-func 17816  df-idfu 17817  df-cofu 17818  df-full 17864  df-fth 17865  df-catc 18057  df-thinc 49908

This theorem is referenced by:  thincciso3  49946  thincciso4  49947