Theorem indthincALT 50390

Description: An alternate proof of indthinc 50389 assuming more axioms including ax-pow 5330 and ax-un 7737. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
indthinc.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
indthinc.h	⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
indthinc.o	⊢ (𝜑 → ∅ = (comp‘𝐶))
indthinc.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
indthincALT	⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Distinct variable groups:   𝑦,𝐵   𝑦,𝐶   𝜑,𝑦

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem indthincALT

Dummy variables 𝑓 𝑔 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		indthinc.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		indthinc.h	. 2 ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
3		1oex 8466	. . . . . 6 ⊢ 1o ∈ V
4	3	ovconst2 7595	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) = 1o)
5		domrefg 8994	. . . . . 6 ⊢ (1o ∈ V → 1o ≼ 1o)
6	3, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7	4, 6	eqbrtrdi 5144	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
8		modom2 9223	. . . 4 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
9	7, 8	sylibr 237	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
10	9	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
11		indthinc.o	. 2 ⊢ (𝜑 → ∅ = (comp‘𝐶))
12		indthinc.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
13		biid 264	. 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))))
14		id 23	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
15	14	ancli 558	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
16	3	ovconst2 7595	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
17		0lt1o 8492	. . . . 5 ⊢ ∅ ∈ 1o
18		eleq2 2849	. . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
19	17, 18	mpbiri 261	. . . 4 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
20	15, 16, 19	3syl 19	. . 3 ⊢ (𝑦 ∈ 𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
21	20	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
22	17	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o)
23		0ov 7451	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
24	23	oveqi 7427	. . . . . 6 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
25		0ov 7451	. . . . . 6 ⊢ (𝑔∅𝑓) = ∅
26	24, 25	eqtri 2783	. . . . 5 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
27	26	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅)
28	3	ovconst2 7595	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
29	28	3adant2 1149	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
30	22, 27, 29	3eltr4d 2875	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
31	30	ad2antrl 741	. 2 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
32	1, 2, 10, 11, 12, 13, 21, 31	isthincd2 50364	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃*wmo 2562  Vcvv 3450  ∅c0 4279  {csn 4584  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  ‘cfv 6533  (class class class)co 7414  1_oc1o 8449   ≼ cdom 8951  Basecbs 17302  Hom chom 17354  compcco 17355  Idccid 17754  ThinCatcthinc 50344

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-1o 8456  df-en 8954  df-dom 8955  df-sdom 8956  df-cat 17757  df-cid 17758  df-thinc 50345

This theorem is used by: (None)