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| Mirrors > Home > MPE Home > Th. List > tfrlem10 | Structured version Visualization version GIF version | ||
| Description: Lemma for transfinite recursion. We define class 𝐶 by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On. Using this assumption we will prove facts about 𝐶 that will lead to a contradiction in tfrlem14 8432, thus showing the domain of recs does in fact equal On. Here we show (under the false assumption) that 𝐶 is a function extending the domain of recs(𝐹) by one. (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)))} | 
| tfrlem.3 | ⊢ 𝐶 = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | 
| Ref | Expression | 
|---|---|
| tfrlem10 | ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvex 6918 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
| 2 | funsng 6616 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
| 3 | 1, 2 | mpan2 691 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | 
| 4 | tfrlem.1 | . . . . . 6 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
| 5 | 4 | tfrlem7 8424 | . . . . 5 ⊢ Fun recs(𝐹) | 
| 6 | 3, 5 | jctil 519 | . . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) | 
| 7 | 1 | dmsnop 6235 | . . . . . 6 ⊢ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉} = {dom recs(𝐹)} | 
| 8 | 7 | ineq2i 4216 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) | 
| 9 | 4 | tfrlem8 8425 | . . . . . 6 ⊢ Ord dom recs(𝐹) | 
| 10 | orddisj 6421 | . . . . . 6 ⊢ (Ord dom recs(𝐹) → (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅) | |
| 11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ (dom recs(𝐹) ∩ {dom recs(𝐹)}) = ∅ | 
| 12 | 8, 11 | eqtri 2764 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅ | 
| 13 | funun 6611 | . . . 4 ⊢ (((Fun recs(𝐹) ∧ Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∧ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅) → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) | |
| 14 | 6, 12, 13 | sylancl 586 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) | 
| 15 | 7 | uneq2i 4164 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | 
| 16 | dmun 5920 | . . . 4 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
| 17 | df-suc 6389 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
| 18 | 15, 16, 17 | 3eqtr4i 2774 | . . 3 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹) | 
| 19 | df-fn 6563 | . . 3 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹) ↔ (Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∧ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹))) | |
| 20 | 14, 18, 19 | sylanblrc 590 | . 2 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) | 
| 21 | tfrlem.3 | . . 3 ⊢ 𝐶 = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
| 22 | 21 | fneq1i 6664 | . 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) | 
| 23 | 20, 22 | sylibr 234 | 1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 {csn 4625 〈cop 4631 dom cdm 5684 ↾ cres 5686 Ord word 6382 Oncon0 6383 suc csuc 6385 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 recscrecs 8411 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 | 
| This theorem is referenced by: tfrlem11 8429 tfrlem12 8430 tfrlem13 8431 | 
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