Theorem indthincALT 49448

Description: An alternate proof of indthinc 49447 assuming more axioms including ax-pow 5304 and ax-un 7671. (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 8398	. . . . . 6 ⊢ 1o ∈ V
4	3	ovconst2 7529	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) = 1o)
5		domrefg 8912	. . . . . 6 ⊢ (1o ∈ V → 1o ≼ 1o)
6	3, 5	ax-mp 5	. . . . 5 ⊢ 1o ≼ 1o
7	4, 6	eqbrtrdi 5131	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
8		modom2 9141	. . . 4 ⊢ (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
9	7, 8	sylibr 234	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
10	9	adantl 481	. 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 261	. 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 22	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵)
15	14	ancli 548	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
16	3	ovconst2 7529	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
17		0lt1o 8422	. . . . 5 ⊢ ∅ ∈ 1o
18		eleq2 2817	. . . . 5 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
19	17, 18	mpbiri 258	. . . 4 ⊢ ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
20	15, 16, 19	3syl 18	. . 3 ⊢ (𝑦 ∈ 𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
21	20	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
22	17	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ 1o)
23		0ov 7386	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩∅𝑧) = ∅
24	23	oveqi 7362	. . . . . 6 ⊢ (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔∅𝑓)
25		0ov 7386	. . . . . 6 ⊢ (𝑔∅𝑓) = ∅
26	24, 25	eqtri 2752	. . . . 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 7529	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
29	28	3adant2 1131	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
30	22, 27, 29	3eltr4d 2843	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
31	30	ad2antrl 728	. 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 49422	1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  Vcvv 3436  ∅c0 4284  {csn 4577  ⟨cop 4583   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617  ‘cfv 6482  (class class class)co 7349  1_oc1o 8381   ≼ cdom 8870  Basecbs 17120  Hom chom 17172  compcco 17173  Idccid 17571  ThinCatcthinc 49402

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-1o 8388  df-en 8873  df-dom 8874  df-sdom 8875  df-cat 17574  df-cid 17575  df-thinc 49403

This theorem is referenced by: (None)