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 17179. (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 2738	. 2 ⊢ (𝜑 → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2738	. 2 ⊢ (𝜑 → (Hom ‘𝑋) = (Hom ‘𝑋))
3		relfull 17871	. . . . . . . . . . . 12 ⊢ Rel (𝑋 Full 𝑌)
4		relin1 5762	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑋) = (Base‘𝑋)
10		eqid 2737	. . . . . . . . . . . . . 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 18072	. . . . . . . . . . . . 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 7989	. . . . . . . . . . 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 588	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
20		eqid 2737	. . . . . . . . . . 11 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
21		eqid 2737	. . . . . . . . . . 11 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
22	9, 20, 21	isffth2 17879	. . . . . . . . . 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 3230	. . . . . . 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 3230	. . . . . 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 7394	. . . . . 6 ⊢ (𝑥(Hom ‘𝑋)𝑦) ∈ V
29	28	f1oen 8913	. . . . 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 17827	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝑋)⟶(Base‘𝑌))
35	34	ffvelcdmda 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
36	35	adantrr 718	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
37	34	ffvelcdmda 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
38	37	adantrl 717	. . . . . 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 9156	. . . . 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 8953	. . . 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 585	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → (𝑥(Hom ‘𝑋)𝑦) ≼ 1o)
44		modom2 9156	. . 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 49569	. 2 ⊢ (𝜑 → 𝑋 ∈ Cat)
47	1, 2, 45, 46	isthincd 49926	1 ⊢ (𝜑 → 𝑋 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052   ∩ cin 3889   class class class wbr 5086  Rel wrel 5630  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  1_oc1o 8392   ≈ cen 8884   ≼ cdom 8885  Basecbs 17173  Hom chom 17225  Isociso 17707   Func cfunc 17815   Full cful 17865   Faith cfth 17866  CatCatccatc 18059  ThinCatcthinc 49907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-struct 17111  df-slot 17146  df-ndx 17158  df-base 17174  df-hom 17238  df-cco 17239  df-cat 17628  df-cid 17629  df-sect 17708  df-inv 17709  df-iso 17710  df-func 17819  df-idfu 17820  df-cofu 17821  df-full 17867  df-fth 17868  df-catc 18060  df-thinc 49908

This theorem is referenced by:  thincciso3  49946  thincciso4  49947