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| Mirrors > Home > MPE Home > Th. List > tfrlem15 | Structured version Visualization version GIF version | ||
| Description: Lemma for transfinite recursion. Without assuming ax-rep 5278, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | 
| Ref | Expression | 
|---|---|
| tfrlem15 | ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tfrlem.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
| 2 | 1 | tfrlem9a 8427 | . . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V) | 
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V) | 
| 4 | 1 | tfrlem13 8431 | . . . 4 ⊢ ¬ recs(𝐹) ∈ V | 
| 5 | simpr 484 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V) | |
| 6 | resss 6018 | . . . . . . . 8 ⊢ (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹) | |
| 7 | 6 | a1i 11 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)) | 
| 8 | 1 | tfrlem6 8423 | . . . . . . . . 9 ⊢ Rel recs(𝐹) | 
| 9 | resdm 6043 | . . . . . . . . 9 ⊢ (Rel recs(𝐹) → (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹)) | |
| 10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ (recs(𝐹) ↾ dom recs(𝐹)) = recs(𝐹) | 
| 11 | ssres2 6021 | . . . . . . . 8 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵)) | |
| 12 | 10, 11 | eqsstrrid 4022 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵)) | 
| 13 | 7, 12 | eqssd 4000 | . . . . . 6 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹)) | 
| 14 | 13 | eleq1d 2825 | . . . . 5 ⊢ (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V)) | 
| 15 | 5, 14 | syl5ibcom 245 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V)) | 
| 16 | 4, 15 | mtoi 199 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵) | 
| 17 | 1 | tfrlem8 8425 | . . . 4 ⊢ Ord dom recs(𝐹) | 
| 18 | eloni 6393 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
| 19 | 18 | adantr 480 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵) | 
| 20 | ordtri1 6416 | . . . . 5 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹))) | |
| 21 | 20 | con2bid 354 | . . . 4 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) | 
| 22 | 17, 19, 21 | sylancr 587 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) | 
| 23 | 16, 22 | mpbird 257 | . 2 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹)) | 
| 24 | 3, 23 | impbida 800 | 1 ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 dom cdm 5684 ↾ cres 5686 Rel wrel 5689 Ord word 6382 Oncon0 6383 Fn wfn 6555 ‘cfv 6560 recscrecs 8411 | 
| 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-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 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-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 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-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 | 
| This theorem is referenced by: tfrlem16 8434 tfr2b 8437 | 
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