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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 7525 | . . 3 ⊢ Ord dom recs(𝐹) |
3 | ordirr 5779 | . . 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 2651 | . . . . 5 ⊢ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
6 | 1, 5 | tfrlem12 7530 | . . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴) |
7 | elssuni 4499 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ ∪ 𝐴) | |
8 | 1 | recsfval 7522 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
9 | 7, 8 | syl6sseqr 3685 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹)) |
10 | dmss 5355 | . . . 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 7139 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V) | |
14 | elon2 5772 | . . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V)) | |
15 | 12, 13, 14 | sylanbrc 699 | . . . . 5 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On) |
16 | sucidg 5841 | . . . . 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 7528 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
19 | fndm 6028 | . . . . 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 2733 | . . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
22 | 11, 21 | sseldd 3637 | . 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 383 = wceq 1523 ∈ wcel 2030 {cab 2637 ∀wral 2941 ∃wrex 2942 Vcvv 3231 ∪ cun 3605 ⊆ wss 3607 {csn 4210 〈cop 4216 ∪ cuni 4468 dom cdm 5143 ↾ cres 5145 Ord word 5760 Oncon0 5761 suc csuc 5763 Fn wfn 5921 ‘cfv 5926 recscrecs 7512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 df-wrecs 7452 df-recs 7513 |
This theorem is referenced by: tfrlem14 7532 tfrlem15 7533 tfrlem16 7534 tfr2b 7537 |
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