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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 5955 | . 2 recs |
3 | 1 | recsfval 5954 | . . . . . . . . 9 recs |
4 | 3 | eleq2i 2145 | . . . . . . . 8 recs |
5 | eluni 3604 | . . . . . . . 8 | |
6 | 4, 5 | bitri 182 | . . . . . . 7 recs |
7 | 3 | eleq2i 2145 | . . . . . . . 8 recs |
8 | eluni 3604 | . . . . . . . 8 | |
9 | 7, 8 | bitri 182 | . . . . . . 7 recs |
10 | 6, 9 | anbi12i 447 | . . . . . 6 recs recs |
11 | eeanv 1848 | . . . . . 6 | |
12 | 10, 11 | bitr4i 185 | . . . . 5 recs recs |
13 | df-br 3786 | . . . . . . . . 9 | |
14 | df-br 3786 | . . . . . . . . 9 | |
15 | 13, 14 | anbi12i 447 | . . . . . . . 8 |
16 | 1 | tfrlem5 5953 | . . . . . . . . 9 |
17 | 16 | impcom 123 | . . . . . . . 8 |
18 | 15, 17 | sylanbr 279 | . . . . . . 7 |
19 | 18 | an4s 552 | . . . . . 6 |
20 | 19 | exlimivv 1817 | . . . . 5 |
21 | 12, 20 | sylbi 119 | . . . 4 recs recs |
22 | 21 | ax-gen 1378 | . . 3 recs recs |
23 | 22 | gen2 1379 | . 2 recs recs |
24 | dffun4 4933 | . 2 recs recs recs recs | |
25 | 2, 23, 24 | mpbir2an 883 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wex 1421 wcel 1433 cab 2067 wral 2348 wrex 2349 cop 3401 cuni 3601 class class class wbr 3785 con0 4118 cres 4365 wrel 4368 wfun 4916 wfn 4917 cfv 4922 recscrecs 5942 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-recs 5943 |
This theorem is referenced by: tfrlem9 5958 tfrlemibfn 5965 tfrlemiubacc 5967 tfri1d 5972 tfrfun 5978 rdgfun 5983 |
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