![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8423 | . . 3 ⊢ Ord dom recs(𝐹) |
3 | ordirr 6404 | . . 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 2735 | . . . . 5 ⊢ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
6 | 1, 5 | tfrlem12 8428 | . . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴) |
7 | elssuni 4942 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ ∪ 𝐴) | |
8 | 1 | recsfval 8420 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
9 | 7, 8 | sseqtrrdi 4047 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹)) |
10 | dmss 5916 | . . . 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 7924 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V) | |
14 | elon2 6397 | . . . . . 6 ⊢ (dom recs(𝐹) ∈ On ↔ (Ord dom recs(𝐹) ∧ dom recs(𝐹) ∈ V)) | |
15 | 12, 13, 14 | sylanbrc 583 | . . . . 5 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ On) |
16 | sucidg 6467 | . . . . 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 8426 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
19 | fndm 6672 | . . . . 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 2842 | . . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
22 | 11, 21 | sseldd 3996 | . 2 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom recs(𝐹)) |
23 | 4, 22 | mto 197 | 1 ⊢ ¬ recs(𝐹) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 {csn 4631 〈cop 4637 ∪ cuni 4912 dom cdm 5689 ↾ cres 5691 Ord word 6385 Oncon0 6386 suc csuc 6388 Fn wfn 6558 ‘cfv 6563 recscrecs 8409 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: tfrlem14 8430 tfrlem15 8431 tfrlem16 8432 tfr2b 8435 |
Copyright terms: Public domain | W3C validator |