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Theorem indthincALT 49135
Description: An alternate proof of indthinc 49134 assuming more axioms including ax-pow 5364 and ax-un 7756. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
indthinc.b (𝜑𝐵 = (Base‘𝐶))
indthinc.h (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
indthinc.o (𝜑 → ∅ = (comp‘𝐶))
indthinc.c (𝜑𝐶𝑉)
Assertion
Ref Expression
indthincALT (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
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.
StepHypRef Expression
1 indthinc.b . 2 (𝜑𝐵 = (Base‘𝐶))
2 indthinc.h . 2 (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))
3 1oex 8517 . . . . . 6 1o ∈ V
43ovconst2 7614 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) = 1o)
5 domrefg 9028 . . . . . 6 (1o ∈ V → 1o ≼ 1o)
63, 5ax-mp 5 . . . . 5 1o ≼ 1o
74, 6eqbrtrdi 5181 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
8 modom2 9282 . . . 4 (∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ↔ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o)
97, 8sylibr 234 . . 3 ((𝑥𝐵𝑦𝐵) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
109adantl 481 . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦))
11 indthinc.o . 2 (𝜑 → ∅ = (comp‘𝐶))
12 indthinc.c . 2 (𝜑𝐶𝑉)
13 biid 261 . 2 (((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))) ↔ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧))))
14 id 22 . . . . 5 (𝑦𝐵𝑦𝐵)
1514ancli 548 . . . 4 (𝑦𝐵 → (𝑦𝐵𝑦𝐵))
163ovconst2 7614 . . . 4 ((𝑦𝐵𝑦𝐵) → (𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o)
17 0lt1o 8543 . . . . 5 ∅ ∈ 1o
18 eleq2 2829 . . . . 5 ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → (∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦) ↔ ∅ ∈ 1o))
1917, 18mpbiri 258 . . . 4 ((𝑦((𝐵 × 𝐵) × {1o})𝑦) = 1o → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
2015, 16, 193syl 18 . . 3 (𝑦𝐵 → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
2120adantl 481 . 2 ((𝜑𝑦𝐵) → ∅ ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑦))
2217a1i 11 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ∅ ∈ 1o)
23 0ov 7469 . . . . . . 7 (⟨𝑥, 𝑦⟩∅𝑧) = ∅
2423oveqi 7445 . . . . . 6 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = (𝑔𝑓)
25 0ov 7469 . . . . . 6 (𝑔𝑓) = ∅
2624, 25eqtri 2764 . . . . 5 (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅
2726a1i 11 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) = ∅)
283ovconst2 7614 . . . . 5 ((𝑥𝐵𝑧𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
29283adant2 1131 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑧) = 1o)
3022, 27, 293eltr4d 2855 . . 3 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
3130ad2antrl 728 . 2 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑓 ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑦) ∧ 𝑔 ∈ (𝑦((𝐵 × 𝐵) × {1o})𝑧)))) → (𝑔(⟨𝑥, 𝑦⟩∅𝑧)𝑓) ∈ (𝑥((𝐵 × 𝐵) × {1o})𝑧))
321, 2, 10, 11, 12, 13, 21, 31isthincd2 49111 1 (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦𝐵 ↦ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  ∃*wmo 2537  Vcvv 3479  c0 4332  {csn 4625  cop 4631   class class class wbr 5142  cmpt 5224   × cxp 5682  cfv 6560  (class class class)co 7432  1oc1o 8500  cdom 8984  Basecbs 17248  Hom chom 17309  compcco 17310  Idccid 17709  ThinCatcthinc 49091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-1o 8507  df-en 8987  df-dom 8988  df-sdom 8989  df-cat 17712  df-cid 17713  df-thinc 49092
This theorem is referenced by: (None)
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