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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 5006, 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 7765 | . . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V) |
3 | 2 | adantl 475 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V) |
4 | 1 | tfrlem13 7769 | . . . 4 ⊢ ¬ recs(𝐹) ∈ V |
5 | simpr 479 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V) | |
6 | resss 5671 | . . . . . . . 8 ⊢ (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)) |
8 | 1 | tfrlem6 7761 | . . . . . . . . 9 ⊢ Rel recs(𝐹) |
9 | resdm 5691 | . . . . . . . . 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 5674 | . . . . . . . 8 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵)) | |
12 | 10, 11 | syl5eqssr 3869 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵)) |
13 | 7, 12 | eqssd 3838 | . . . . . 6 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹)) |
14 | 13 | eleq1d 2844 | . . . . 5 ⊢ (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V)) |
15 | 5, 14 | syl5ibcom 237 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V)) |
16 | 4, 15 | mtoi 191 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵) |
17 | 1 | tfrlem8 7763 | . . . 4 ⊢ Ord dom recs(𝐹) |
18 | eloni 5986 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
19 | 18 | adantr 474 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵) |
20 | ordtri1 6009 | . . . . 5 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹))) | |
21 | 20 | con2bid 346 | . . . 4 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) |
22 | 17, 19, 21 | sylancr 581 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) |
23 | 16, 22 | mpbird 249 | . 2 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹)) |
24 | 3, 23 | impbida 791 | 1 ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {cab 2763 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ⊆ wss 3792 dom cdm 5355 ↾ cres 5357 Rel wrel 5360 Ord word 5975 Oncon0 5976 Fn wfn 6130 ‘cfv 6135 recscrecs 7750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-wrecs 7689 df-recs 7751 |
This theorem is referenced by: tfrlem16 7772 tfr2b 7775 |
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