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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 5192, 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 8024 | . . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹) ↾ 𝐵) ∈ V) |
3 | 2 | adantl 484 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ dom recs(𝐹)) → (recs(𝐹) ↾ 𝐵) ∈ V) |
4 | 1 | tfrlem13 8028 | . . . 4 ⊢ ¬ recs(𝐹) ∈ V |
5 | simpr 487 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V) | |
6 | resss 5880 | . . . . . . . 8 ⊢ (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)) |
8 | 1 | tfrlem6 8020 | . . . . . . . . 9 ⊢ Rel recs(𝐹) |
9 | resdm 5899 | . . . . . . . . 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 5883 | . . . . . . . 8 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵)) | |
12 | 10, 11 | eqsstrrid 4018 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵)) |
13 | 7, 12 | eqssd 3986 | . . . . . 6 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹)) |
14 | 13 | eleq1d 2899 | . . . . 5 ⊢ (dom recs(𝐹) ⊆ 𝐵 → ((recs(𝐹) ↾ 𝐵) ∈ V ↔ recs(𝐹) ∈ V)) |
15 | 5, 14 | syl5ibcom 247 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ∈ V)) |
16 | 4, 15 | mtoi 201 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → ¬ dom recs(𝐹) ⊆ 𝐵) |
17 | 1 | tfrlem8 8022 | . . . 4 ⊢ Ord dom recs(𝐹) |
18 | eloni 6203 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
19 | 18 | adantr 483 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵) |
20 | ordtri1 6226 | . . . . 5 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (dom recs(𝐹) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ dom recs(𝐹))) | |
21 | 20 | con2bid 357 | . . . 4 ⊢ ((Ord dom recs(𝐹) ∧ Ord 𝐵) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) |
22 | 17, 19, 21 | sylancr 589 | . . 3 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (𝐵 ∈ dom recs(𝐹) ↔ ¬ dom recs(𝐹) ⊆ 𝐵)) |
23 | 16, 22 | mpbird 259 | . 2 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → 𝐵 ∈ dom recs(𝐹)) |
24 | 3, 23 | impbida 799 | 1 ⊢ (𝐵 ∈ On → (𝐵 ∈ dom recs(𝐹) ↔ (recs(𝐹) ↾ 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 dom cdm 5557 ↾ cres 5559 Rel wrel 5562 Ord word 6192 Oncon0 6193 Fn wfn 6352 ‘cfv 6357 recscrecs 8009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-wrecs 7949 df-recs 8010 |
This theorem is referenced by: tfrlem16 8031 tfr2b 8034 |
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