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Mirrors > Home > MPE Home > Th. List > tfrlem13 | Structured version Visualization version GIF version |
Description: Lemma for transfinite recursion. If recs is a set function, then 𝐶 is acceptable, and thus a subset of recs, but dom 𝐶 is bigger than dom recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem13 | ⊢ ¬ recs(𝐹) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem8 7480 | . . 3 ⊢ Ord dom recs(𝐹) |
3 | ordirr 5741 | . . 3 ⊢ (Ord dom recs(𝐹) → ¬ dom recs(𝐹) ∈ dom recs(𝐹)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ¬ dom recs(𝐹) ∈ dom recs(𝐹) |
5 | eqid 2622 | . . . . 5 ⊢ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
6 | 1, 5 | tfrlem12 7485 | . . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴) |
7 | elssuni 4467 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ ∪ 𝐴) | |
8 | 1 | recsfval 7477 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
9 | 7, 8 | syl6sseqr 3652 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹)) |
10 | dmss 5323 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹) → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ dom recs(𝐹)) | |
11 | 6, 9, 10 | 3syl 18 | . . 3 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ dom recs(𝐹)) |
12 | 2 | a1i 11 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → Ord dom recs(𝐹)) |
13 | dmexg 7097 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V) | |
14 | elon2 5734 | . . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V)) | |
15 | 12, 13, 14 | sylanbrc 698 | . . . . 5 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On) |
16 | sucidg 5803 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → dom recs(𝐹) ∈ suc dom recs(𝐹)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ suc dom recs(𝐹)) |
18 | 1, 5 | tfrlem10 7483 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
19 | fndm 5990 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹) → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹)) | |
20 | 15, 18, 19 | 3syl 18 | . . . 4 ⊢ (recs(𝐹) ∈ V → dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = suc dom recs(𝐹)) |
21 | 17, 20 | eleqtrrd 2704 | . . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
22 | 11, 21 | sseldd 3604 | . 2 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹)) |
23 | 4, 22 | mto 188 | 1 ⊢ ¬ recs(𝐹) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 {csn 4177 〈cop 4183 ∪ cuni 4436 dom cdm 5114 ↾ cres 5116 Ord word 5722 Oncon0 5723 suc csuc 5725 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: tfrlem14 7487 tfrlem15 7488 tfrlem16 7489 tfr2b 7492 |
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