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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 8422 | . 2 ⊢ Rel recs(𝐹) | 
| 3 | 1 | recsfval 8421 | . . . . . . . . 9 ⊢ recs(𝐹) = ∪ 𝐴 | 
| 4 | 3 | eleq2i 2833 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐴) | 
| 5 | eluni 4910 | . . . . . . . 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 2833 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ recs(𝐹) ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐴) | 
| 8 | eluni 4910 | . . . . . . . 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 1955 | . . . . . 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 5144 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
| 14 | df-br 5144 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
| 15 | 13, 14 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 〈𝑥, 𝑣〉 ∈ ℎ)) | 
| 16 | 1 | tfrlem5 8420 | . . . . . . . . 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 1932 | . . . . 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 1795 | . . 3 ⊢ ∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) | 
| 23 | 22 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑢∀𝑣((〈𝑥, 𝑢〉 ∈ recs(𝐹) ∧ 〈𝑥, 𝑣〉 ∈ recs(𝐹)) → 𝑢 = 𝑣) | 
| 24 | dffun4 6577 | . 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 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 〈cop 4632 ∪ cuni 4907 class class class wbr 5143 ↾ cres 5687 Rel wrel 5690 Oncon0 6384 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 recscrecs 8410 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 | 
| This theorem is referenced by: tfrlem9 8425 tfrlem9a 8426 tfrlem10 8427 tfrlem14 8431 tfrlem16 8433 tfr1a 8434 tfr1 8437 | 
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