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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 6201 | . . . . . . . 8 |
3 | 2 | abeq2i 2248 | . . . . . . 7 |
4 | fndm 5217 | . . . . . . . . . . 11 | |
5 | 4 | adantr 274 | . . . . . . . . . 10 |
6 | 5 | eleq1d 2206 | . . . . . . . . 9 |
7 | 6 | biimprcd 159 | . . . . . . . 8 |
8 | 7 | rexlimiv 2541 | . . . . . . 7 |
9 | 3, 8 | sylbi 120 | . . . . . 6 |
10 | eleq1a 2209 | . . . . . 6 | |
11 | 9, 10 | syl 14 | . . . . 5 |
12 | 11 | rexlimiv 2541 | . . . 4 |
13 | 12 | abssi 3167 | . . 3 |
14 | ssorduni 4398 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 1 | recsfval 6205 | . . . . 5 recs |
17 | 16 | dmeqi 4735 | . . . 4 recs |
18 | dmuni 4744 | . . . 4 | |
19 | vex 2684 | . . . . . 6 | |
20 | 19 | dmex 4800 | . . . . 5 |
21 | 20 | dfiun2 3842 | . . . 4 |
22 | 17, 18, 21 | 3eqtri 2162 | . . 3 recs |
23 | ordeq 4289 | . . 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 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 wss 3066 cuni 3731 ciun 3808 word 4279 con0 4280 cdm 4534 cres 4536 wfn 5113 cfv 5118 recscrecs 6194 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-tr 4022 df-iord 4283 df-on 4285 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-recs 6195 |
This theorem is referenced by: tfrlemi14d 6223 tfri1dALT 6241 |
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