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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 5932 | . 2 recs |
3 | 1 | recsfval 5931 | . . . . . . . . 9 recs |
4 | 3 | eleq2i 2104 | . . . . . . . 8 recs |
5 | eluni 3583 | . . . . . . . 8 | |
6 | 4, 5 | bitri 173 | . . . . . . 7 recs |
7 | 3 | eleq2i 2104 | . . . . . . . 8 recs |
8 | eluni 3583 | . . . . . . . 8 | |
9 | 7, 8 | bitri 173 | . . . . . . 7 recs |
10 | 6, 9 | anbi12i 433 | . . . . . 6 recs recs |
11 | eeanv 1807 | . . . . . 6 | |
12 | 10, 11 | bitr4i 176 | . . . . 5 recs recs |
13 | df-br 3765 | . . . . . . . . 9 | |
14 | df-br 3765 | . . . . . . . . 9 | |
15 | 13, 14 | anbi12i 433 | . . . . . . . 8 |
16 | 1 | tfrlem5 5930 | . . . . . . . . 9 |
17 | 16 | impcom 116 | . . . . . . . 8 |
18 | 15, 17 | sylanbr 269 | . . . . . . 7 |
19 | 18 | an4s 522 | . . . . . 6 |
20 | 19 | exlimivv 1776 | . . . . 5 |
21 | 12, 20 | sylbi 114 | . . . 4 recs recs |
22 | 21 | ax-gen 1338 | . . 3 recs recs |
23 | 22 | gen2 1339 | . 2 recs recs |
24 | dffun4 4913 | . 2 recs recs recs recs | |
25 | 2, 23, 24 | mpbir2an 849 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wex 1381 wcel 1393 cab 2026 wral 2306 wrex 2307 cop 3378 cuni 3580 class class class wbr 3764 con0 4100 cres 4347 wrel 4350 wfun 4896 wfn 4897 cfv 4902 recscrecs 5919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 |
This theorem is referenced by: tfrlem9 5935 tfrlemibfn 5942 tfrlemiubacc 5944 tfri1d 5949 tfrfun 5955 rdgfun 5960 |
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