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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 8421 | . 2 ⊢ Rel recs(𝐹) |
3 | 1 | recsfval 8420 | . . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴 |
4 | 3 | eleq2i 2831 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐴) |
5 | eluni 4915 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐴 ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) | |
6 | 4, 5 | bitri 275 | . . . . . . 7 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) |
7 | 3 | eleq2i 2831 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐴) |
8 | eluni 4915 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐴 ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) | |
9 | 7, 8 | bitri 275 | . . . . . . 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 1953 | . . . . . 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 278 | . . . . 5 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
13 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
14 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
15 | 13, 14 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ)) |
16 | 1 | tfrlem5 8419 | . . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
17 | 16 | impcom 407 | . . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
18 | 15, 17 | sylanbr 582 | . . . . . . 7 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
19 | 18 | an4s 660 | . . . . . 6 ⊢ (((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
20 | 19 | exlimivv 1930 | . . . . 5 ⊢ (∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
21 | 12, 20 | sylbi 217 | . . . 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 6579 | . 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 711 | 1 ⊢ Fun recs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 〈cop 4637 ∪ cuni 4912 class class class wbr 5148 ↾ cres 5691 Rel wrel 5694 Oncon0 6386 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 recscrecs 8409 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: tfrlem9 8424 tfrlem9a 8425 tfrlem10 8426 tfrlem14 8430 tfrlem16 8432 tfr1a 8433 tfr1 8436 |
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