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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 5275 | . . . 4 | |
3 | simpll 519 | . . . . . . 7 | |
4 | simpr 109 | . . . . . . 7 | |
5 | 3, 4 | fveq12d 5487 | . . . . . 6 |
6 | 3, 4 | reseq12d 4879 | . . . . . . 7 |
7 | 6 | fveq2d 5484 | . . . . . 6 |
8 | 5, 7 | eqeq12d 2179 | . . . . 5 |
9 | simpr 109 | . . . . . 6 | |
10 | 9 | adantr 274 | . . . . 5 |
11 | 8, 10 | cbvraldva2 2696 | . . . 4 |
12 | 2, 11 | anbi12d 465 | . . 3 |
13 | 12 | cbvrexdva 2699 | . 2 |
14 | tfrlem3.1 | . 2 | |
15 | 1, 13, 14 | elab2 2869 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 cab 2150 wral 2442 wrex 2443 cvv 2721 con0 4335 cres 4600 wfn 5177 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: tfrlem3 6270 tfrlem5 6273 tfrlemisucaccv 6284 tfrlemibxssdm 6286 tfrlemi14d 6292 tfrexlem 6293 |
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