Theorem thincciso2 49487

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 17121. (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 2732	. 2 ⊢ (𝜑 → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2732	. 2 ⊢ (𝜑 → (Hom ‘𝑋) = (Hom ‘𝑋))
3		relfull 17812	. . . . . . . . . . . 12 ⊢ Rel (𝑋 Full 𝑌)
4		relin1 5747	. . . . . . . . . . . 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 2731	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑋) = (Base‘𝑋)
10		eqid 2731	. . . . . . . . . . . . . 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 18013	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
16	6, 15	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
17	16	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
18		1st2ndbr 7969	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
20		eqid 2731	. . . . . . . . . . 11 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
21		eqid 2731	. . . . . . . . . . 11 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
22	9, 20, 21	isffth2 17820	. . . . . . . . . 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 218	. . . . . . . . 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 495	. . . . . . . 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 3224	. . . . . . 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 3224	. . . . . 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 466	. . . . 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 7374	. . . . . 6 ⊢ (𝑥(Hom ‘𝑋)𝑦) ∈ V
29	28	f1oen 8890	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → 𝑌 ∈ ThinCat)
33	23	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
34	9, 10, 33	funcf1 17768	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝑋)⟶(Base‘𝑌))
35	34	ffvelcdmda 7012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
36	35	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
37	34	ffvelcdmda 7012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
38	37	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
39	32, 36, 38, 10, 21	thincmo 49460	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)))
40		modom2 9131	. . . . 5 ⊢ (∃*𝑓 𝑓 ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ↔ (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
41	39, 40	sylib 218	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (((1st ‘𝐹)‘𝑥)(Hom ‘𝑌)((1st ‘𝐹)‘𝑦)) ≼ 1o)
42		endomtr 8929	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
44		modom2 9131	. . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦) ↔ (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
45	43, 44	sylibr 234	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦))
46	33	funcrcl2 49111	. 2 ⊢ (𝜑 → 𝑋 ∈ Cat)
47	1, 2, 45, 46	isthincd 49468	1 ⊢ (𝜑 → 𝑋 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533  ∀wral 3047   ∩ cin 3896   class class class wbr 5086  Rel wrel 5616  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  1_oc1o 8373   ≈ cen 8861   ≼ cdom 8862  Basecbs 17115  Hom chom 17167  Isociso 17648   Func cfunc 17756   Full cful 17806   Faith cfth 17807  CatCatccatc 18000  ThinCatcthinc 49449

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  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 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-struct 17053  df-slot 17088  df-ndx 17100  df-base 17116  df-hom 17180  df-cco 17181  df-cat 17569  df-cid 17570  df-sect 17649  df-inv 17650  df-iso 17651  df-func 17760  df-idfu 17761  df-cofu 17762  df-full 17808  df-fth 17809  df-catc 18001  df-thinc 49450

This theorem is referenced by:  thincciso3  49488  thincciso4  49489