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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 8026, 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 6682 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
2 | funsng 6404 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
3 | 1, 2 | mpan2 689 | . . . . 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 8018 | . . . . 5 ⊢ Fun recs(𝐹) |
6 | 3, 5 | jctil 522 | . . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
7 | 1 | dmsnop 6072 | . . . . . 6 ⊢ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉} = {dom recs(𝐹)} |
8 | 7 | ineq2i 4185 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) |
9 | 4 | tfrlem8 8019 | . . . . . 6 ⊢ Ord dom recs(𝐹) |
10 | orddisj 6228 | . . . . . 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 2844 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅ |
13 | funun 6399 | . . . 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 588 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
15 | 7 | uneq2i 4135 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) |
16 | dmun 5778 | . . . 4 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
17 | df-suc 6196 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
18 | 15, 16, 17 | 3eqtr4i 2854 | . . 3 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹) |
19 | df-fn 6357 | . . 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 592 | . 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 6449 | . 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
23 | 20, 22 | sylibr 236 | 1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3933 ∩ cin 3934 ∅c0 4290 {csn 4566 〈cop 4572 dom cdm 5554 ↾ cres 5556 Ord word 6189 Oncon0 6190 suc csuc 6192 Fun wfun 6348 Fn wfn 6349 ‘cfv 6354 recscrecs 8006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 df-wrecs 7946 df-recs 8007 |
This theorem is referenced by: tfrlem11 8023 tfrlem12 8024 tfrlem13 8025 |
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