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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 5285, 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 8425 | . . 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 8429 | . . . 4 ⊢ ¬ recs(𝐹) ∈ V |
5 | simpr 484 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → (recs(𝐹) ↾ 𝐵) ∈ V) | |
6 | resss 6022 | . . . . . . . 8 ⊢ (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) ⊆ recs(𝐹)) |
8 | 1 | tfrlem6 8421 | . . . . . . . . 9 ⊢ Rel recs(𝐹) |
9 | resdm 6046 | . . . . . . . . 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 6025 | . . . . . . . 8 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ dom recs(𝐹)) ⊆ (recs(𝐹) ↾ 𝐵)) | |
12 | 10, 11 | eqsstrrid 4045 | . . . . . . 7 ⊢ (dom recs(𝐹) ⊆ 𝐵 → recs(𝐹) ⊆ (recs(𝐹) ↾ 𝐵)) |
13 | 7, 12 | eqssd 4013 | . . . . . 6 ⊢ (dom recs(𝐹) ⊆ 𝐵 → (recs(𝐹) ↾ 𝐵) = recs(𝐹)) |
14 | 13 | eleq1d 2824 | . . . . 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 8423 | . . . 4 ⊢ Ord dom recs(𝐹) |
18 | eloni 6396 | . . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝐵 ∈ On ∧ (recs(𝐹) ↾ 𝐵) ∈ V) → Ord 𝐵) |
20 | ordtri1 6419 | . . . . 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 801 | 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 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 dom cdm 5689 ↾ cres 5691 Rel wrel 5694 Ord word 6385 Oncon0 6386 Fn wfn 6558 ‘cfv 6563 recscrecs 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: tfrlem16 8432 tfr2b 8435 |
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