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Mirrors > Home > ILE Home > Th. List > tfrcllemubacc | Unicode version |
Description: Lemma for tfrcl 6254. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f | recs |
tfrcl.g | |
tfrcl.x | |
tfrcl.ex | |
tfrcllemsucfn.1 | |
tfrcllembacc.3 | |
tfrcllembacc.u | |
tfrcllembacc.4 | |
tfrcllembacc.5 |
Ref | Expression |
---|---|
tfrcllemubacc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcl.f | . . . . . . . . 9 recs | |
2 | tfrcl.g | . . . . . . . . 9 | |
3 | tfrcl.x | . . . . . . . . 9 | |
4 | tfrcl.ex | . . . . . . . . 9 | |
5 | tfrcllemsucfn.1 | . . . . . . . . 9 | |
6 | tfrcllembacc.3 | . . . . . . . . 9 | |
7 | tfrcllembacc.u | . . . . . . . . 9 | |
8 | tfrcllembacc.4 | . . . . . . . . 9 | |
9 | tfrcllembacc.5 | . . . . . . . . 9 | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfrcllembfn 6247 | . . . . . . . 8 |
11 | fdm 5273 | . . . . . . . 8 | |
12 | 10, 11 | syl 14 | . . . . . . 7 |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfrcllembacc 6245 | . . . . . . . . . 10 |
14 | 13 | unissd 3755 | . . . . . . . . 9 |
15 | 5, 3 | tfrcllemssrecs 6242 | . . . . . . . . 9 recs |
16 | 14, 15 | sstrd 3102 | . . . . . . . 8 recs |
17 | dmss 4733 | . . . . . . . 8 recs recs | |
18 | 16, 17 | syl 14 | . . . . . . 7 recs |
19 | 12, 18 | eqsstrrd 3129 | . . . . . 6 recs |
20 | 19 | sselda 3092 | . . . . 5 recs |
21 | eqid 2137 | . . . . . 6 | |
22 | 21 | tfrlem9 6209 | . . . . 5 recs recs recs |
23 | 20, 22 | syl 14 | . . . 4 recs recs |
24 | tfrfun 6210 | . . . . 5 recs | |
25 | 12 | eleq2d 2207 | . . . . . 6 |
26 | 25 | biimpar 295 | . . . . 5 |
27 | funssfv 5440 | . . . . 5 recs recs recs | |
28 | 24, 16, 26, 27 | mp3an2ani 1322 | . . . 4 recs |
29 | ordelon 4300 | . . . . . . . . . 10 | |
30 | 3, 8, 29 | syl2anc 408 | . . . . . . . . 9 |
31 | eloni 4292 | . . . . . . . . 9 | |
32 | 30, 31 | syl 14 | . . . . . . . 8 |
33 | ordelss 4296 | . . . . . . . 8 | |
34 | 32, 33 | sylan 281 | . . . . . . 7 |
35 | 12 | adantr 274 | . . . . . . 7 |
36 | 34, 35 | sseqtrrd 3131 | . . . . . 6 |
37 | fun2ssres 5161 | . . . . . 6 recs recs recs | |
38 | 24, 16, 36, 37 | mp3an2ani 1322 | . . . . 5 recs |
39 | 38 | fveq2d 5418 | . . . 4 recs |
40 | 23, 28, 39 | 3eqtr3d 2178 | . . 3 |
41 | 40 | ralrimiva 2503 | . 2 |
42 | fveq2 5414 | . . . 4 | |
43 | reseq2 4809 | . . . . 5 | |
44 | 43 | fveq2d 5418 | . . . 4 |
45 | 42, 44 | eqeq12d 2152 | . . 3 |
46 | 45 | cbvralv 2652 | . 2 |
47 | 41, 46 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wex 1468 wcel 1480 cab 2123 wral 2414 wrex 2415 cun 3064 wss 3066 csn 3522 cop 3525 cuni 3731 word 4279 con0 4280 csuc 4282 cdm 4534 cres 4536 wfun 5112 wfn 5113 wf 5114 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-in1 603 ax-in2 604 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 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 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-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 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-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-recs 6195 |
This theorem is referenced by: tfrcllemex 6250 |
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