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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 5154, 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 8009 | . . 3 ⊢ ¬ recs(𝐹) ∈ V |
3 | 1 | tfrlem7 8002 | . . . 4 ⊢ Fun recs(𝐹) |
4 | funex 6959 | . . . 4 ⊢ ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V) | |
5 | 3, 4 | mpan 689 | . . 3 ⊢ (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V) |
6 | 2, 5 | mto 200 | . 2 ⊢ ¬ dom recs(𝐹) ∈ On |
7 | 1 | tfrlem8 8003 | . . 3 ⊢ Ord dom recs(𝐹) |
8 | ordeleqon 7483 | . . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)) | |
9 | 7, 8 | mpbi 233 | . 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On) |
10 | 6, 9 | mtpor 1772 | 1 ⊢ dom recs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 Vcvv 3441 dom cdm 5519 ↾ cres 5521 Ord word 6158 Oncon0 6159 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 recscrecs 7990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-wrecs 7930 df-recs 7991 |
This theorem is referenced by: tfr1 8016 |
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