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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 8386, 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 6894 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
2 | funsng 6589 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
3 | 1, 2 | mpan2 688 | . . . . 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 8378 | . . . . 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 6205 | . . . . . 6 ⊢ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉} = {dom recs(𝐹)} |
8 | 7 | ineq2i 4201 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) |
9 | 4 | tfrlem8 8379 | . . . . . 6 ⊢ Ord dom recs(𝐹) |
10 | orddisj 6392 | . . . . . 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 2752 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅ |
13 | funun 6584 | . . . 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 585 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
15 | 7 | uneq2i 4152 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) |
16 | dmun 5900 | . . . 4 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
17 | df-suc 6360 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
18 | 15, 16, 17 | 3eqtr4i 2762 | . . 3 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹) |
19 | df-fn 6536 | . . 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 589 | . 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 6636 | . 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
23 | 20, 22 | sylibr 233 | 1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2701 ∀wral 3053 ∃wrex 3062 Vcvv 3466 ∪ cun 3938 ∩ cin 3939 ∅c0 4314 {csn 4620 〈cop 4626 dom cdm 5666 ↾ cres 5668 Ord word 6353 Oncon0 6354 suc csuc 6356 Fun wfun 6527 Fn wfn 6528 ‘cfv 6533 recscrecs 8365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-ov 7404 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 |
This theorem is referenced by: tfrlem11 8383 tfrlem12 8384 tfrlem13 8385 |
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