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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 5182, 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 8020 | . . 3 ⊢ ¬ recs(𝐹) ∈ V |
3 | 1 | tfrlem7 8013 | . . . 4 ⊢ Fun recs(𝐹) |
4 | funex 6976 | . . . 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 199 | . 2 ⊢ ¬ dom recs(𝐹) ∈ On |
7 | 1 | tfrlem8 8014 | . . 3 ⊢ Ord dom recs(𝐹) |
8 | ordeleqon 7497 | . . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)) | |
9 | 7, 8 | mpbi 232 | . 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On) |
10 | 6, 9 | mtpor 1767 | 1 ⊢ dom recs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 dom cdm 5549 ↾ cres 5551 Ord word 6184 Oncon0 6185 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 recscrecs 8001 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-wrecs 7941 df-recs 8002 |
This theorem is referenced by: tfr1 8027 |
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