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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 8424 | . . 3 ⊢ Ord dom recs(𝐹) |
| 3 | ordirr 6402 | . . 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 2737 | . . . . 5 ⊢ (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) = (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) | |
| 6 | 1, 5 | tfrlem12 8429 | . . . 4 ⊢ (recs(𝐹) ∈ V → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴) |
| 7 | elssuni 4937 | . . . . 5 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ ∪ 𝐴) | |
| 8 | 1 | recsfval 8421 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
| 9 | 7, 8 | sseqtrrdi 4025 | . . . 4 ⊢ ((recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ∈ 𝐴 → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) ⊆ recs(𝐹)) |
| 10 | dmss 5913 | . . . 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 7923 | . . . . . 6 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ V) | |
| 14 | elon2 6395 | . . . . . 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 6465 | . . . . 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 8427 | . . . . 5 ⊢ (dom recs(𝐹) ∈ On → (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉}) Fn suc dom recs(𝐹)) |
| 19 | fndm 6671 | . . . . 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 2844 | . . 3 ⊢ (recs(𝐹) ∈ V → dom recs(𝐹) ∈ dom (recs(𝐹) ∪ {〈dom recs(𝐹), (𝐹‘recs(𝐹))〉})) |
| 22 | 11, 21 | sseldd 3984 | . 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 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 〈cop 4632 ∪ cuni 4907 dom cdm 5685 ↾ cres 5687 Ord word 6383 Oncon0 6384 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 recscrecs 8410 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 |
| This theorem is referenced by: tfrlem14 8431 tfrlem15 8432 tfrlem16 8433 tfr2b 8436 |
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