Theorem thincciso2 50117

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 17274. (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 2770	. 2 ⊢ (𝜑 → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2770	. 2 ⊢ (𝜑 → (Hom ‘𝑋) = (Hom ‘𝑋))
3		relfull 17966	. . . . . . . . . . . 12 ⊢ Rel (𝑋 Full 𝑌)
4		relin1 5800	. . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑋) = (Base‘𝑋)
10		eqid 2769	. . . . . . . . . . . . . 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 18167	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
16	6, 15	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
17	16	simpld 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
18		1st2ndbr 8038	. . . . . . . . . . 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 598	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
20		eqid 2769	. . . . . . . . . . 11 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
21		eqid 2769	. . . . . . . . . . 11 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
22	9, 20, 21	isffth2 17974	. . . . . . . . . 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 221	. . . . . . . . 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 500	. . . . . . . 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 3263	. . . . . . 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 3263	. . . . . 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 471	. . . . 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 7444	. . . . . 6 ⊢ (𝑥(Hom ‘𝑋)𝑦) ∈ V
29	28	f1oen 8968	. . . . 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 18	. . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → 𝑌 ∈ ThinCat)
33	23	simpld 499	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
34	9, 10, 33	funcf1 17922	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝑋)⟶(Base‘𝑌))
35	34	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
36	35	adantrr 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
37	34	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
38	37	adantrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
39	32, 36, 38, 10, 21	thincmo 50090	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
40		modom2 9211	. . . . 5 ⊢ (∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ↔ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
41	39, 40	sylib 221	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
42		endomtr 9008	. . . 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 595	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
44		modom2 9211	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦) ↔ (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
45	43, 44	sylibr 237	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦))
46	33	funcrcl2 49741	. 2 ⊢ (𝜑 → 𝑋 ∈ Cat)
47	1, 2, 45, 46	isthincd 50098	1 ⊢ (𝜑 → 𝑋 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571  ∀wral 3085   ∩ cin 3912   class class class wbr 5113  Rel wrel 5667  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  1_oc1o 8445   ≈ cen 8939   ≼ cdom 8940  Basecbs 17268  Hom chom 17320  Isociso 17802   Func cfunc 17910   Full cful 17960   Faith cfth 17961  CatCatccatc 18154  ThinCatcthinc 50079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-cat 17723  df-cid 17724  df-sect 17803  df-inv 17804  df-iso 17805  df-func 17914  df-idfu 17915  df-cofu 17916  df-full 17962  df-fth 17963  df-catc 18155  df-thinc 50080

This theorem is referenced by:  thincciso3  50118  thincciso4  50119