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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 5008, 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 7771 | . . 3 ⊢ ¬ recs(𝐹) ∈ V |
3 | 1 | tfrlem7 7764 | . . . 4 ⊢ Fun recs(𝐹) |
4 | funex 6756 | . . . 4 ⊢ ((Fun recs(𝐹) ∧ dom recs(𝐹) ∈ On) → recs(𝐹) ∈ V) | |
5 | 3, 4 | mpan 680 | . . 3 ⊢ (dom recs(𝐹) ∈ On → recs(𝐹) ∈ V) |
6 | 2, 5 | mto 189 | . 2 ⊢ ¬ dom recs(𝐹) ∈ On |
7 | 1 | tfrlem8 7765 | . . 3 ⊢ Ord dom recs(𝐹) |
8 | ordeleqon 7268 | . . 3 ⊢ (Ord dom recs(𝐹) ↔ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On)) | |
9 | 7, 8 | mpbi 222 | . 2 ⊢ (dom recs(𝐹) ∈ On ∨ dom recs(𝐹) = On) |
10 | 6, 9 | mtpor 1814 | 1 ⊢ dom recs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 {cab 2763 ∀wral 3090 ∃wrex 3091 Vcvv 3398 dom cdm 5357 ↾ cres 5359 Ord word 5977 Oncon0 5978 Fun wfun 6131 Fn wfn 6132 ‘cfv 6137 recscrecs 7752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-wrecs 7691 df-recs 7753 |
This theorem is referenced by: tfr1 7778 |
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