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| Mirrors > Home > MPE Home > Th. List > tfrlem8 | Structured version Visualization version GIF version | ||
| Description: Lemma for transfinite recursion. The domain of recs is an ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) | 
| Ref | Expression | 
|---|---|
| tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | 
| Ref | Expression | 
|---|---|
| tfrlem8 | ⊢ Ord dom recs(𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tfrlem.1 | . . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
| 2 | 1 | tfrlem3 8418 | . . . . . . . 8 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))} | 
| 3 | 2 | eqabri 2885 | . . . . . . 7 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) | 
| 4 | fndm 6671 | . . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧) | |
| 5 | 4 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 = 𝑧) | 
| 6 | 5 | eleq1d 2826 | . . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On)) | 
| 7 | 6 | biimprcd 250 | . . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On)) | 
| 8 | 7 | rexlimiv 3148 | . . . . . . 7 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On) | 
| 9 | 3, 8 | sylbi 217 | . . . . . 6 ⊢ (𝑔 ∈ 𝐴 → dom 𝑔 ∈ On) | 
| 10 | eleq1a 2836 | . . . . . 6 ⊢ (dom 𝑔 ∈ On → (𝑧 = dom 𝑔 → 𝑧 ∈ On)) | |
| 11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝑔 ∈ 𝐴 → (𝑧 = dom 𝑔 → 𝑧 ∈ On)) | 
| 12 | 11 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔 → 𝑧 ∈ On) | 
| 13 | 12 | abssi 4070 | . . 3 ⊢ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On | 
| 14 | ssorduni 7799 | . . 3 ⊢ ({𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On → Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}) | |
| 15 | 13, 14 | ax-mp 5 | . 2 ⊢ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} | 
| 16 | 1 | recsfval 8421 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 | 
| 17 | 16 | dmeqi 5915 | . . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴 | 
| 18 | dmuni 5925 | . . . 4 ⊢ dom ∪ 𝐴 = ∪ 𝑔 ∈ 𝐴 dom 𝑔 | |
| 19 | vex 3484 | . . . . . 6 ⊢ 𝑔 ∈ V | |
| 20 | 19 | dmex 7931 | . . . . 5 ⊢ dom 𝑔 ∈ V | 
| 21 | 20 | dfiun2 5033 | . . . 4 ⊢ ∪ 𝑔 ∈ 𝐴 dom 𝑔 = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} | 
| 22 | 17, 18, 21 | 3eqtri 2769 | . . 3 ⊢ dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} | 
| 23 | ordeq 6391 | . . 3 ⊢ (dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} → (Ord dom recs(𝐹) ↔ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔})) | |
| 24 | 22, 23 | ax-mp 5 | . 2 ⊢ (Ord dom recs(𝐹) ↔ Ord ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔}) | 
| 25 | 15, 24 | mpbir 231 | 1 ⊢ Ord dom recs(𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 ∪ cuni 4907 ∪ ciun 4991 dom cdm 5685 ↾ cres 5687 Ord word 6383 Oncon0 6384 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-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-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: tfrlem10 8427 tfrlem12 8429 tfrlem13 8430 tfrlem14 8431 tfrlem15 8432 tfrlem16 8433 | 
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