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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 6295 | . 2 recs |
3 | 1 | recsfval 6294 | . . . . . . . . 9 recs |
4 | 3 | eleq2i 2237 | . . . . . . . 8 recs |
5 | eluni 3799 | . . . . . . . 8 | |
6 | 4, 5 | bitri 183 | . . . . . . 7 recs |
7 | 3 | eleq2i 2237 | . . . . . . . 8 recs |
8 | eluni 3799 | . . . . . . . 8 | |
9 | 7, 8 | bitri 183 | . . . . . . 7 recs |
10 | 6, 9 | anbi12i 457 | . . . . . 6 recs recs |
11 | eeanv 1925 | . . . . . 6 | |
12 | 10, 11 | bitr4i 186 | . . . . 5 recs recs |
13 | df-br 3990 | . . . . . . . . 9 | |
14 | df-br 3990 | . . . . . . . . 9 | |
15 | 13, 14 | anbi12i 457 | . . . . . . . 8 |
16 | 1 | tfrlem5 6293 | . . . . . . . . 9 |
17 | 16 | impcom 124 | . . . . . . . 8 |
18 | 15, 17 | sylanbr 283 | . . . . . . 7 |
19 | 18 | an4s 583 | . . . . . 6 |
20 | 19 | exlimivv 1889 | . . . . 5 |
21 | 12, 20 | sylbi 120 | . . . 4 recs recs |
22 | 21 | ax-gen 1442 | . . 3 recs recs |
23 | 22 | gen2 1443 | . 2 recs recs |
24 | dffun4 5209 | . 2 recs recs recs recs | |
25 | 2, 23, 24 | mpbir2an 937 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wceq 1348 wex 1485 wcel 2141 cab 2156 wral 2448 wrex 2449 cop 3586 cuni 3796 class class class wbr 3989 con0 4348 cres 4613 wrel 4616 wfun 5192 wfn 5193 cfv 5198 recscrecs 6283 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 df-recs 6284 |
This theorem is referenced by: tfrlem9 6298 tfrfun 6299 tfrlemibfn 6307 tfrlemiubacc 6309 tfri1d 6314 rdgfun 6352 |
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