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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 7818 | . . . . . . . 8 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))} |
3 | 2 | abeq2i 2900 | . . . . . . 7 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) |
4 | fndm 6288 | . . . . . . . . . . 11 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧) | |
5 | 4 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 = 𝑧) |
6 | 5 | eleq1d 2850 | . . . . . . . . 9 ⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → (dom 𝑔 ∈ On ↔ 𝑧 ∈ On)) |
7 | 6 | biimprcd 242 | . . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On)) |
8 | 7 | rexlimiv 3225 | . . . . . . 7 ⊢ (∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))) → dom 𝑔 ∈ On) |
9 | 3, 8 | sylbi 209 | . . . . . 6 ⊢ (𝑔 ∈ 𝐴 → dom 𝑔 ∈ On) |
10 | eleq1a 2861 | . . . . . 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 3225 | . . . 4 ⊢ (∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔 → 𝑧 ∈ On) |
13 | 12 | abssi 3936 | . . 3 ⊢ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} ⊆ On |
14 | ssorduni 7316 | . . 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 7821 | . . . . 5 ⊢ recs(𝐹) = ∪ 𝐴 |
17 | 16 | dmeqi 5623 | . . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴 |
18 | dmuni 5632 | . . . 4 ⊢ dom ∪ 𝐴 = ∪ 𝑔 ∈ 𝐴 dom 𝑔 | |
19 | vex 3418 | . . . . . 6 ⊢ 𝑔 ∈ V | |
20 | 19 | dmex 7431 | . . . . 5 ⊢ dom 𝑔 ∈ V |
21 | 20 | dfiun2 4828 | . . . 4 ⊢ ∪ 𝑔 ∈ 𝐴 dom 𝑔 = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} |
22 | 17, 18, 21 | 3eqtri 2806 | . . 3 ⊢ dom recs(𝐹) = ∪ {𝑧 ∣ ∃𝑔 ∈ 𝐴 𝑧 = dom 𝑔} |
23 | ordeq 6036 | . . 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 223 | 1 ⊢ Ord dom recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 {cab 2758 ∀wral 3088 ∃wrex 3089 ⊆ wss 3829 ∪ cuni 4712 ∪ ciun 4792 dom cdm 5407 ↾ cres 5409 Ord word 6028 Oncon0 6029 Fn wfn 6183 ‘cfv 6188 recscrecs 7811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-tr 5031 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-iota 6152 df-fun 6190 df-fn 6191 df-fv 6196 df-wrecs 7750 df-recs 7812 |
This theorem is referenced by: tfrlem10 7827 tfrlem12 7829 tfrlem13 7830 tfrlem14 7831 tfrlem15 7832 tfrlem16 7833 |
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