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Mirrors > Home > MPE Home > Th. List > tfrlem7 | Structured version Visualization version GIF version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfrlem.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
Ref | Expression |
---|---|
tfrlem7 | ⊢ Fun recs(𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . 3 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem6 8012 | . 2 ⊢ Rel recs(𝐹) |
3 | 1 | recsfval 8011 | . . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴 |
4 | 3 | eleq2i 2904 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐴) |
5 | eluni 4834 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐴 ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) | |
6 | 4, 5 | bitri 277 | . . . . . . 7 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) |
7 | 3 | eleq2i 2904 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐴) |
8 | eluni 4834 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐴 ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) | |
9 | 7, 8 | bitri 277 | . . . . . . 7 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) |
10 | 6, 9 | anbi12i 628 | . . . . . 6 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ (∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
11 | exdistrv 1952 | . . . . . 6 ⊢ (∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) ↔ (∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) | |
12 | 10, 11 | bitr4i 280 | . . . . 5 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
13 | df-br 5059 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
14 | df-br 5059 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
15 | 13, 14 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ)) |
16 | 1 | tfrlem5 8010 | . . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
17 | 16 | impcom 410 | . . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
18 | 15, 17 | sylanbr 584 | . . . . . . 7 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
19 | 18 | an4s 658 | . . . . . 6 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
20 | 19 | exlimivv 1929 | . . . . 5 ⊢ (∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
21 | 12, 20 | sylbi 219 | . . . 4 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
22 | 21 | ax-gen 1792 | . . 3 ⊢ ∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
23 | 22 | gen2 1793 | . 2 ⊢ ∀𝑥∀𝑢∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
24 | dffun4 6361 | . 2 ⊢ (Fun recs(𝐹) ↔ (Rel recs(𝐹) ∧ ∀𝑥∀𝑢∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣))) | |
25 | 2, 23, 24 | mpbir2an 709 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 〈cop 4566 ∪ cuni 4831 class class class wbr 5058 ↾ cres 5551 Rel wrel 5554 Oncon0 6185 Fun wfun 6343 Fn wfn 6344 ‘cfv 6349 recscrecs 8001 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-wrecs 7941 df-recs 8002 |
This theorem is referenced by: tfrlem9 8015 tfrlem9a 8016 tfrlem10 8017 tfrlem14 8021 tfrlem16 8023 tfr1a 8024 tfr1 8027 |
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