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Mirrors > Home > MPE Home > Th. List > tfrlem13 | Structured version Visualization version GIF version |
Description: Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised 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 |
---|---|
tfrlem13 | ⊢ ¬ recs(𝐹) ∈ 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 | tfrlem8 7634 | . . 3 ⊢ Ord dom recs(𝐹) |
3 | ordirr 5885 | . . 3 ⊢ (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ¬ dom recs(𝐹) ∈ dom recs(𝐹) |
5 | eqid 2771 | . . . . 5 ⊢ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
6 | 1, 5 | tfrlem12 7639 | . . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴) |
7 | elssuni 4604 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ ∪ 𝐴) | |
8 | 1 | recsfval 7631 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
9 | 7, 8 | syl6sseqr 3802 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹)) |
10 | dmss 5462 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹) → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ dom recs(𝐹)) | |
11 | 6, 9, 10 | 3syl 18 | . . 3 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ dom recs(𝐹)) |
12 | 2 | a1i 11 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹)) |
13 | dmexg 7245 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V) | |
14 | elon2 5878 | . . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V)) | |
15 | 12, 13, 14 | sylanbrc 566 | . . . . 5 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On) |
16 | sucidg 5947 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ suc dom recs(𝐹)) |
18 | 1, 5 | tfrlem10 7637 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
19 | fndm 6131 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹) → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹)) | |
20 | 15, 18, 19 | 3syl 18 | . . . 4 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹)) |
21 | 17, 20 | eleqtrrd 2853 | . . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
22 | 11, 21 | sseldd 3754 | . 2 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹)) |
23 | 4, 22 | mto 188 | 1 ⊢ ¬ recs(𝐹) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {cab 2757 ∀wral 3061 ∃wrex 3062 Vcvv 3351 ∪ cun 3722 ⊆ wss 3724 {csn 4317 〈cop 4323 ∪ cuni 4575 dom cdm 5250 ↾ cres 5252 Ord word 5866 Oncon0 5867 suc csuc 5869 Fn wfn 6027 ‘cfv 6032 recscrecs 7621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-fv 6040 df-wrecs 7560 df-recs 7622 |
This theorem is referenced by: tfrlem14 7641 tfrlem15 7642 tfrlem16 7643 tfr2b 7646 |
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