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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 8391, 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 6905 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
2 | funsng 6600 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) | |
3 | 1, 2 | mpan2 690 | . . . . 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 8383 | . . . . 5 ⊢ Fun recs(𝐹) |
6 | 3, 5 | jctil 521 | . . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})) |
7 | 1 | dmsnop 6216 | . . . . . 6 ⊢ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩} = {dom recs(𝐹)} |
8 | 7 | ineq2i 4210 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) |
9 | 4 | tfrlem8 8384 | . . . . . 6 ⊢ Ord dom recs(𝐹) |
10 | orddisj 6403 | . . . . . 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 2761 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = ∅ |
13 | funun 6595 | . . . 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 587 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩})) |
15 | 7 | uneq2i 4161 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) |
16 | dmun 5911 | . . . 4 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = (dom recs(𝐹) ∪ dom {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) | |
17 | df-suc 6371 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
18 | 15, 16, 17 | 3eqtr4i 2771 | . . 3 ⊢ dom (recs(𝐹) ∪ {⟨dom recs(𝐹), (𝐹‘recs(𝐹))⟩}) = suc dom recs(𝐹) |
19 | df-fn 6547 | . . 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 591 | . 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 6647 | . 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 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 ∅c0 4323 {csn 4629 ⟨cop 4635 dom cdm 5677 ↾ cres 5679 Ord word 6364 Oncon0 6365 suc csuc 6367 Fun wfun 6538 Fn wfn 6539 ‘cfv 6544 recscrecs 8370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-ov 7412 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 |
This theorem is referenced by: tfrlem11 8388 tfrlem12 8389 tfrlem13 8390 |
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