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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 8412 | . 2 ⊢ Rel recs(𝐹) |
3 | 1 | recsfval 8411 | . . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴 |
4 | 3 | eleq2i 2818 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐴) |
5 | eluni 4916 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐴 ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) | |
6 | 4, 5 | bitri 274 | . . . . . . 7 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴)) |
7 | 3 | eleq2i 2818 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐴) |
8 | eluni 4916 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐴 ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) | |
9 | 7, 8 | bitri 274 | . . . . . . 7 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) |
10 | 6, 9 | anbi12i 626 | . . . . . 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 277 | . . . . 5 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) ↔ ∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴))) |
13 | df-br 5154 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
14 | df-br 5154 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
15 | 13, 14 | anbi12i 626 | . . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ)) |
16 | 1 | tfrlem5 8410 | . . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
17 | 16 | impcom 406 | . . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
18 | 15, 17 | sylanbr 580 | . . . . . . 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 1928 | . . . . 5 ⊢ (∃𝑔∃ℎ((〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐴) ∧ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐴)) → 𝑢 = 𝑣) |
21 | 12, 20 | sylbi 216 | . . . 4 ⊢ ((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
22 | 21 | ax-gen 1790 | . . 3 ⊢ ∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
23 | 22 | gen2 1791 | . 2 ⊢ ∀𝑥∀𝑢∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) |
24 | dffun4 6570 | . 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 394 ∀wal 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2703 ∀wral 3051 ∃wrex 3060 〈cop 4639 ∪ cuni 4913 class class class wbr 5153 ↾ cres 5684 Rel wrel 5687 Oncon0 6376 Fun wfun 6548 Fn wfn 6549 ‘cfv 6554 recscrecs 8400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fo 6560 df-fv 6562 df-ov 7427 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 |
This theorem is referenced by: tfrlem9 8415 tfrlem9a 8416 tfrlem10 8417 tfrlem14 8421 tfrlem16 8423 tfr1a 8424 tfr1 8427 |
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