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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 7523 | . 2 ⊢ Rel recs(𝐹) |
3 | 1 | recsfval 7522 | . . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴 |
4 | 3 | eleq2i 2722 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐴) |
5 | eluni 4471 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐴 ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) | |
6 | 4, 5 | bitri 264 | . . . . . . 7 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) |
7 | 3 | eleq2i 2722 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐴) |
8 | eluni 4471 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐴 ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) | |
9 | 7, 8 | bitri 264 | . . . . . . 7 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) |
10 | 6, 9 | anbi12i 733 | . . . . . 6 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ (∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
11 | eeanv 2218 | . . . . . 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 267 | . . . . 5 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
13 | df-br 4686 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
14 | df-br 4686 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
15 | 13, 14 | anbi12i 733 | . . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ)) |
16 | 1 | tfrlem5 7521 | . . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
17 | 16 | impcom 445 | . . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
18 | 15, 17 | sylanbr 489 | . . . . . . 7 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
19 | 18 | an4s 886 | . . . . . 6 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
20 | 19 | exlimivv 1900 | . . . . 5 ⊢ (∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
21 | 12, 20 | sylbi 207 | . . . 4 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
22 | 21 | ax-gen 1762 | . . 3 ⊢ ∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
23 | 22 | gen2 1763 | . 2 ⊢ ∀𝑥∀𝑢∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
24 | dffun4 5938 | . 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 975 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1521 = wceq 1523 ∃wex 1744 ∈ wcel 2030 {cab 2637 ∀wral 2941 ∃wrex 2942 〈cop 4216 ∪ cuni 4468 class class class wbr 4685 ↾ cres 5145 Rel wrel 5148 Oncon0 5761 Fun wfun 5920 Fn wfn 5921 ‘cfv 5926 recscrecs 7512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 df-wrecs 7452 df-recs 7513 |
This theorem is referenced by: tfrlem9 7526 tfrlem9a 7527 tfrlem10 7528 tfrlem14 7532 tfrlem16 7534 tfr1a 7535 tfr1 7538 |
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