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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 7487, 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 6201 | . . . . . 6 ⊢ (𝐹‘recs(𝐹)) ∈ V | |
2 | funsng 5937 | . . . . . 6 ⊢ ((dom recs(𝐹) ∈ On ∧ (𝐹‘recs(𝐹)) ∈ V) → Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
3 | 1, 2 | mpan2 707 | . . . . 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 7479 | . . . . 5 ⊢ Fun recs(𝐹) |
6 | 3, 5 | jctil 560 | . . . 4 ⊢ (dom recs(𝐹) ∈ On → (Fun recs(𝐹) ∧ Fun {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
7 | 1 | dmsnop 5609 | . . . . . 6 ⊢ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉} = {dom recs(𝐹)} |
8 | 7 | ineq2i 3811 | . . . . 5 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∩ {dom recs(𝐹)}) |
9 | 4 | tfrlem8 7480 | . . . . . 6 ⊢ Ord dom recs(𝐹) |
10 | orddisj 5762 | . . . . . 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 2644 | . . . 4 ⊢ (dom recs(𝐹) ∩ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = ∅ |
13 | funun 5932 | . . . 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 694 | . . 3 ⊢ (dom recs(𝐹) ∈ On → Fun (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
15 | 7 | uneq2i 3764 | . . . 4 ⊢ (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) |
16 | dmun 5331 | . . . 4 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (dom recs(𝐹) ∪ dom {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
17 | df-suc 5729 | . . . 4 ⊢ suc dom recs(𝐹) = (dom recs(𝐹) ∪ {dom recs(𝐹)}) | |
18 | 15, 16, 17 | 3eqtr4i 2654 | . . 3 ⊢ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹) |
19 | df-fn 5891 | . . 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 697 | . 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 5985 | . 2 ⊢ (𝐶 Fn suc dom recs(𝐹) ↔ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
23 | 20, 22 | sylibr 224 | 1 ⊢ (dom recs(𝐹) ∈ On → 𝐶 Fn suc dom recs(𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 〈cop 4183 dom cdm 5114 ↾ cres 5116 Ord word 5722 Oncon0 5723 suc csuc 5725 Fun wfun 5882 Fn wfn 5883 ‘cfv 5888 recscrecs 7467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-wrecs 7407 df-recs 7468 |
This theorem is referenced by: tfrlem11 7484 tfrlem12 7485 tfrlem13 7486 |
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