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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 4992 | . . . 4 | |
3 | simpll 481 | . . . . . . 7 | |
4 | simpr 103 | . . . . . . 7 | |
5 | 3, 4 | fveq12d 5184 | . . . . . 6 |
6 | 3, 4 | reseq12d 4613 | . . . . . . 7 |
7 | 6 | fveq2d 5182 | . . . . . 6 |
8 | 5, 7 | eqeq12d 2054 | . . . . 5 |
9 | simpr 103 | . . . . . 6 | |
10 | 9 | adantr 261 | . . . . 5 |
11 | 8, 10 | cbvraldva2 2537 | . . . 4 |
12 | 2, 11 | anbi12d 442 | . . 3 |
13 | 12 | cbvrexdva 2540 | . 2 |
14 | tfrlem3.1 | . 2 | |
15 | 1, 13, 14 | elab2 2690 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wcel 1393 cab 2026 wral 2306 wrex 2307 cvv 2557 con0 4100 cres 4347 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: tfrlem3 5926 tfrlem5 5930 tfrlemisucaccv 5939 tfrlemibxssdm 5941 tfrlemi14d 5947 tfrexlem 5948 |
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