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Mirrors > Home > MPE Home > Th. List > tfrlem14 | Structured version Visualization version GIF version |
Description: Lemma for transfinite recursion. Assuming ax-rep 5289, dom recs ∈ V ↔ recs ∈ V, so since dom recs is an ordinal, it must be equal to On. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem14 | ⊢ dom recs(𝐹) = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem13 8419 | . . 3 ⊢ ¬ recs(𝐹) ∈ V |
3 | 1 | tfrlem7 8412 | . . . 4 ⊢ Fun recs(𝐹) |
4 | funex 7237 | . . . 4 ⊢ ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V) | |
5 | 3, 4 | mpan 688 | . . 3 ⊢ (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V) |
6 | 2, 5 | mto 196 | . 2 ⊢ ¬ dom recs(𝐹) ∈ On |
7 | 1 | tfrlem8 8413 | . . 3 ⊢ Ord dom recs(𝐹) |
8 | ordeleqon 7792 | . . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)) | |
9 | 7, 8 | mpbi 229 | . 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On) |
10 | 6, 9 | mtpor 1764 | 1 ⊢ dom recs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 {cab 2705 ∀wral 3058 ∃wrex 3067 Vcvv 3473 dom cdm 5682 ↾ cres 5684 Ord word 6373 Oncon0 6374 Fun wfun 6547 Fn wfn 6548 ‘cfv 6553 recscrecs 8399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 |
This theorem is referenced by: tfr1 8426 |
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