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Mirrors > Home > ILE Home > Th. List > tfrlem8 | Unicode version |
Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
Ref | Expression |
---|---|
tfrlem.1 |
Ref | Expression |
---|---|
tfrlem8 | recs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . . . . . . . 9 | |
2 | 1 | tfrlem3 6176 | . . . . . . . 8 |
3 | 2 | abeq2i 2228 | . . . . . . 7 |
4 | fndm 5192 | . . . . . . . . . . 11 | |
5 | 4 | adantr 274 | . . . . . . . . . 10 |
6 | 5 | eleq1d 2186 | . . . . . . . . 9 |
7 | 6 | biimprcd 159 | . . . . . . . 8 |
8 | 7 | rexlimiv 2520 | . . . . . . 7 |
9 | 3, 8 | sylbi 120 | . . . . . 6 |
10 | eleq1a 2189 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | rexlimiv 2520 | . . . 4 |
13 | 12 | abssi 3142 | . . 3 |
14 | ssorduni 4373 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 1 | recsfval 6180 | . . . . 5 recs |
17 | 16 | dmeqi 4710 | . . . 4 recs |
18 | dmuni 4719 | . . . 4 | |
19 | vex 2663 | . . . . . 6 | |
20 | 19 | dmex 4775 | . . . . 5 |
21 | 20 | dfiun2 3817 | . . . 4 |
22 | 17, 18, 21 | 3eqtri 2142 | . . 3 recs |
23 | ordeq 4264 | . . 3 recs recs | |
24 | 22, 23 | ax-mp 5 | . 2 recs |
25 | 15, 24 | mpbir 145 | 1 recs |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 wss 3041 cuni 3706 ciun 3783 word 4254 con0 4255 cdm 4509 cres 4511 wfn 5088 cfv 5093 recscrecs 6169 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-tr 3997 df-iord 4258 df-on 4260 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-recs 6170 |
This theorem is referenced by: tfrlemi14d 6198 tfri1dALT 6216 |
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