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Mirrors > Home > ILE Home > Th. List > tfrlem3a | Unicode version |
Description: Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Ref | Expression |
---|---|
tfrlem3.1 | |
tfrlem3.2 |
Ref | Expression |
---|---|
tfrlem3a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem3.2 | . 2 | |
2 | fneq12 5186 | . . . 4 | |
3 | simpll 503 | . . . . . . 7 | |
4 | simpr 109 | . . . . . . 7 | |
5 | 3, 4 | fveq12d 5396 | . . . . . 6 |
6 | 3, 4 | reseq12d 4790 | . . . . . . 7 |
7 | 6 | fveq2d 5393 | . . . . . 6 |
8 | 5, 7 | eqeq12d 2132 | . . . . 5 |
9 | simpr 109 | . . . . . 6 | |
10 | 9 | adantr 274 | . . . . 5 |
11 | 8, 10 | cbvraldva2 2635 | . . . 4 |
12 | 2, 11 | anbi12d 464 | . . 3 |
13 | 12 | cbvrexdva 2638 | . 2 |
14 | tfrlem3.1 | . 2 | |
15 | 1, 13, 14 | elab2 2805 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 cvv 2660 con0 4255 cres 4511 wfn 5088 cfv 5093 |
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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 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-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: tfrlem3 6176 tfrlem5 6179 tfrlemisucaccv 6190 tfrlemibxssdm 6192 tfrlemi14d 6198 tfrexlem 6199 |
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