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Mirrors > Home > ILE Home > Th. List > tfrlem7 | Unicode 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 |
Ref | Expression |
---|---|
tfrlem7 | recs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . 3 | |
2 | 1 | tfrlem6 6263 | . 2 recs |
3 | 1 | recsfval 6262 | . . . . . . . . 9 recs |
4 | 3 | eleq2i 2224 | . . . . . . . 8 recs |
5 | eluni 3775 | . . . . . . . 8 | |
6 | 4, 5 | bitri 183 | . . . . . . 7 recs |
7 | 3 | eleq2i 2224 | . . . . . . . 8 recs |
8 | eluni 3775 | . . . . . . . 8 | |
9 | 7, 8 | bitri 183 | . . . . . . 7 recs |
10 | 6, 9 | anbi12i 456 | . . . . . 6 recs recs |
11 | eeanv 1912 | . . . . . 6 | |
12 | 10, 11 | bitr4i 186 | . . . . 5 recs recs |
13 | df-br 3966 | . . . . . . . . 9 | |
14 | df-br 3966 | . . . . . . . . 9 | |
15 | 13, 14 | anbi12i 456 | . . . . . . . 8 |
16 | 1 | tfrlem5 6261 | . . . . . . . . 9 |
17 | 16 | impcom 124 | . . . . . . . 8 |
18 | 15, 17 | sylanbr 283 | . . . . . . 7 |
19 | 18 | an4s 578 | . . . . . 6 |
20 | 19 | exlimivv 1876 | . . . . 5 |
21 | 12, 20 | sylbi 120 | . . . 4 recs recs |
22 | 21 | ax-gen 1429 | . . 3 recs recs |
23 | 22 | gen2 1430 | . 2 recs recs |
24 | dffun4 5181 | . 2 recs recs recs recs | |
25 | 2, 23, 24 | mpbir2an 927 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1333 wceq 1335 wex 1472 wcel 2128 cab 2143 wral 2435 wrex 2436 cop 3563 cuni 3772 class class class wbr 3965 con0 4323 cres 4588 wrel 4591 wfun 5164 wfn 5165 cfv 5170 recscrecs 6251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-res 4598 df-iota 5135 df-fun 5172 df-fn 5173 df-fv 5178 df-recs 6252 |
This theorem is referenced by: tfrlem9 6266 tfrfun 6267 tfrlemibfn 6275 tfrlemiubacc 6277 tfri1d 6282 rdgfun 6320 |
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