Theorem thincciso2 49959

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 17180. (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 2742	. 2 ⊢ (𝜑 → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2742	. 2 ⊢ (𝜑 → (Hom ‘𝑋) = (Hom ‘𝑋))
3		relfull 17872	. . . . . . . . . . . 12 ⊢ Rel (𝑋 Full 𝑌)
4		relin1 5758	. . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑋) = (Base‘𝑋)
10		eqid 2741	. . . . . . . . . . . . . 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 18073	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌))))
16	6, 15	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝐹):(Base‘𝑋)–1-1-onto→(Base‘𝑌)))
17	16	simpld 496	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
18		1st2ndbr 7988	. . . . . . . . . . 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 594	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))(2nd ‘𝐹))
20		eqid 2741	. . . . . . . . . . 11 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
21		eqid 2741	. . . . . . . . . . 11 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
22	9, 20, 21	isffth2 17880	. . . . . . . . . 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 220	. . . . . . . . 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 497	. . . . . . . 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 3233	. . . . . . 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 3233	. . . . . 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 468	. . . . 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 7393	. . . . . 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 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → 𝑌 ∈ ThinCat)
33	23	simpld 496	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹)(𝑋 Func 𝑌)(2nd ‘𝐹))
34	9, 10, 33	funcf1 17828	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝑋)⟶(Base‘𝑌))
35	34	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
36	35	adantrr 724	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝑌))
37	34	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝑋)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
38	37	adantrl 723	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝑌))
39	32, 36, 38, 10, 21	thincmo 49932	. . . . 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 220	. . . 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 591	. . 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 236	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑋)𝑦))
46	33	funcrcl2 49583	. 2 ⊢ (𝜑 → 𝑋 ∈ Cat)
47	1, 2, 45, 46	isthincd 49940	1 ⊢ (𝜑 → 𝑋 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃*wmo 2543  ∀wral 3055   ∩ cin 3884   class class class wbr 5075  Rel wrel 5626  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  1_oc1o 8392   ≈ cen 8884   ≼ cdom 8885  Basecbs 17174  Hom chom 17226  Isociso 17708   Func cfunc 17816   Full cful 17866   Faith cfth 17867  CatCatccatc 18060  ThinCatcthinc 49921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  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 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  df-cat 17629  df-cid 17630  df-sect 17709  df-inv 17710  df-iso 17711  df-func 17820  df-idfu 17821  df-cofu 17822  df-full 17868  df-fth 17869  df-catc 18061  df-thinc 49922

This theorem is referenced by:  thincciso3  49960  thincciso4  49961