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| Mirrors > Home > MPE Home > Th. List > Mathboxes > indthincALT | Structured version Visualization version GIF version | ||
| 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.) | 
| 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) | 
| Ref | Expression | 
|---|---|
| indthincALT | ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | indthinc.b | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶)) | |
| 2 | indthinc.h | . 2 ⊢ (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶)) | |
| 3 | 1oex 8517 | . . . . . 6 ⊢ 1o ∈ V | |
| 4 | 3 | ovconst2 7614 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) = 1o) | 
| 5 | domrefg 9028 | . . . . . 6 ⊢ (1o ∈ V → 1o ≼ 1o) | |
| 6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ 1o ≼ 1o | 
| 7 | 4, 6 | eqbrtrdi 5181 | . . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥((𝐵 × 𝐵) × {1o})𝑦) ≼ 1o) | 
| 8 | modom2 9282 | . . . 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 7614 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (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)) | |
| 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 7469 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉∅𝑧) = ∅ | |
| 24 | 23 | oveqi 7445 | . . . . . 6 ⊢ (𝑔(〈𝑥, 𝑦〉∅𝑧)𝑓) = (𝑔∅𝑓) | 
| 25 | 0ov 7469 | . . . . . 6 ⊢ (𝑔∅𝑓) = ∅ | |
| 26 | 24, 25 | eqtri 2764 | . . . . 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 7614 | . . . . 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 2855 | . . 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 49111 | 1 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅))) | 
| 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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