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Theorem isthincd2 49086
Description: The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthincd.b (𝜑𝐵 = (Base‘𝐶))
isthincd.h (𝜑𝐻 = (Hom ‘𝐶))
isthincd.t ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
isthincd2.o (𝜑· = (comp‘𝐶))
isthincd2.c (𝜑𝐶𝑉)
isthincd2.ps (𝜓 ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))
isthincd2.1 ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))
isthincd2.2 ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
Assertion
Ref Expression
isthincd2 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
Distinct variable groups:   𝑦,𝐵   𝐶,𝑓,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦   1 ,𝑓,𝑔,𝑥,𝑧   · ,𝑓,𝑔,𝑥,𝑦,𝑧   𝐵,𝑓,𝑔,𝑥,𝑧   𝐶,𝑔,𝑧   𝑓,𝐻,𝑔,𝑥,𝑦,𝑧   𝜑,𝑔,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑔)   1 (𝑦)   𝑉(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem isthincd2
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isthincd.b . . 3 (𝜑𝐵 = (Base‘𝐶))
2 isthincd.h . . 3 (𝜑𝐻 = (Hom ‘𝐶))
3 isthincd.t . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
4 isthincd2.o . . . . 5 (𝜑· = (comp‘𝐶))
5 isthincd2.c . . . . 5 (𝜑𝐶𝑉)
6 3an4anass 1105 . . . . . . . 8 (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ 𝑤𝐵) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)))
76anbi1i 624 . . . . . . 7 ((((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ 𝑤𝐵) ∧ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
8 isthincd2.ps . . . . . . . . 9 (𝜓 ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))
983anbi1i 1158 . . . . . . . 8 ((𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)) ↔ (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) ∧ 𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)))
10 3anass 1095 . . . . . . . 8 ((((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) ∧ 𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)) ↔ (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) ∧ (𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))))
11 an4 656 . . . . . . . 8 ((((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) ∧ (𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ 𝑤𝐵) ∧ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
129, 10, 113bitri 297 . . . . . . 7 ((𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)) ↔ (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ 𝑤𝐵) ∧ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
13 df-3an 1089 . . . . . . . 8 ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))
1413anbi2i 623 . . . . . . 7 ((((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧)) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
157, 12, 143bitr4i 303 . . . . . 6 ((𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
16 df-3an 1089 . . . . . 6 (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ (((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
1715, 16bitr4i 278 . . . . 5 ((𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
18 isthincd2.1 . . . . 5 ((𝜑𝑦𝐵) → 1 ∈ (𝑦𝐻𝑦))
19 simpr1l 1231 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑥𝐵)
20 simpr1r 1232 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑦𝐵)
21 simpr31 1264 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑓 ∈ (𝑥𝐻𝑦))
2220, 18syldan 591 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 1 ∈ (𝑦𝐻𝑦))
238bianass 642 . . . . . . . . . . . 12 ((𝜑𝜓) ↔ ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))
24 isthincd2.2 . . . . . . . . . . . 12 ((𝜑𝜓) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2523, 24sylbir 235 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2625ralrimivva 3202 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2726ralrimivvva 3205 . . . . . . . . 9 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2827adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ∀𝑥𝐵𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
2919, 20, 20, 21, 22, 28isthincd2lem2 49084 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) ∈ (𝑥𝐻𝑦))
303ralrimivva 3202 . . . . . . . 8 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
3130adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
3219, 20, 29, 21, 31isthincd2lem1 49075 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
3317, 32sylan2b 594 . . . . 5 ((𝜑 ∧ (𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))) → ( 1 (⟨𝑥, 𝑦· 𝑦)𝑓) = 𝑓)
34 simpr2l 1233 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑧𝐵)
35 simpr32 1265 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑔 ∈ (𝑦𝐻𝑧))
3620, 20, 34, 22, 35, 28isthincd2lem2 49084 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) ∈ (𝑦𝐻𝑧))
3720, 34, 36, 35, 31isthincd2lem1 49075 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
3817, 37sylan2b 594 . . . . 5 ((𝜑 ∧ (𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))) → (𝑔(⟨𝑦, 𝑦· 𝑧) 1 ) = 𝑔)
39243ad2antr1 1189 . . . . 5 ((𝜑 ∧ (𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
40 simpr2r 1234 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑤𝐵)
41 simpr33 1266 . . . . . . . . 9 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑘 ∈ (𝑧𝐻𝑤))
4220, 34, 40, 35, 41, 28isthincd2lem2 49084 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑦, 𝑧· 𝑤)𝑔) ∈ (𝑦𝐻𝑤))
4319, 20, 40, 21, 42, 28isthincd2lem2 49084 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) ∈ (𝑥𝐻𝑤))
4417, 39sylan2br 595 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
4519, 34, 40, 44, 41, 28isthincd2lem2 49084 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)) ∈ (𝑥𝐻𝑤))
4619, 40, 43, 45, 31isthincd2lem1 49075 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝑧𝐵𝑤𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
4717, 46sylan2b 594 . . . . 5 ((𝜑 ∧ (𝜓𝑤𝐵𝑘 ∈ (𝑧𝐻𝑤))) → ((𝑘(⟨𝑦, 𝑧· 𝑤)𝑔)(⟨𝑥, 𝑦· 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧· 𝑤)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓)))
481, 2, 4, 5, 17, 18, 33, 38, 39, 47iscatd2 17724 . . . 4 (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
4948simpld 494 . . 3 (𝜑𝐶 ∈ Cat)
501, 2, 3, 49isthincd 49085 . 2 (𝜑𝐶 ∈ ThinCat)
5148simprd 495 . 2 (𝜑 → (Id‘𝐶) = (𝑦𝐵1 ))
5250, 51jca 511 1 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  cop 4632  cmpt 5225  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  ThinCatcthinc 49067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-cat 17711  df-cid 17712  df-thinc 49068
This theorem is referenced by:  indthinc  49109  indthincALT  49110  prsthinc  49111
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