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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 7770, 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 6459 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
2 | funsng 6185 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
3 | 1, 2 | mpan2 681 | . . . . 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 7762 | . . . . 5 ⊢ Fun recs(𝐹) |
6 | 3, 5 | jctil 515 | . . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
7 | 1 | dmsnop 5863 | . . . . . 6 ⊢ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉} = {dom recs(𝐹)} |
8 | 7 | ineq2i 4033 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) |
9 | 4 | tfrlem8 7763 | . . . . . 6 ⊢ Ord dom recs(𝐹) |
10 | orddisj 6014 | . . . . . 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 2801 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅ |
13 | funun 6180 | . . . 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 580 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
15 | 7 | uneq2i 3986 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) |
16 | dmun 5576 | . . . 4 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
17 | df-suc 5982 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
18 | 15, 16, 17 | 3eqtr4i 2811 | . . 3 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹) |
19 | df-fn 6138 | . . 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 584 | . 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 6230 | . 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
23 | 20, 22 | sylibr 226 | 1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {cab 2762 ∀wral 3089 ∃wrex 3090 Vcvv 3397 ∪ cun 3789 ∩ cin 3790 ∅c0 4140 {csn 4397 〈cop 4403 dom cdm 5355 ↾ cres 5357 Ord word 5975 Oncon0 5976 suc csuc 5978 Fun wfun 6129 Fn wfn 6130 ‘cfv 6135 recscrecs 7750 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-wrecs 7689 df-recs 7751 |
This theorem is referenced by: tfrlem11 7767 tfrlem12 7768 tfrlem13 7769 |
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