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Mirrors > Home > ILE Home > Th. List > tfr1onlemubacc | Unicode version |
Description: Lemma for tfr1on 6247. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
Ref | Expression |
---|---|
tfr1on.f | recs |
tfr1on.g | |
tfr1on.x | |
tfr1on.ex | |
tfr1onlemsucfn.1 | |
tfr1onlembacc.3 | |
tfr1onlembacc.u | |
tfr1onlembacc.4 | |
tfr1onlembacc.5 |
Ref | Expression |
---|---|
tfr1onlemubacc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr1on.f | . . . . . . . . 9 recs | |
2 | tfr1on.g | . . . . . . . . 9 | |
3 | tfr1on.x | . . . . . . . . 9 | |
4 | tfr1on.ex | . . . . . . . . 9 | |
5 | tfr1onlemsucfn.1 | . . . . . . . . 9 | |
6 | tfr1onlembacc.3 | . . . . . . . . 9 | |
7 | tfr1onlembacc.u | . . . . . . . . 9 | |
8 | tfr1onlembacc.4 | . . . . . . . . 9 | |
9 | tfr1onlembacc.5 | . . . . . . . . 9 | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlembfn 6241 | . . . . . . . 8 |
11 | fndm 5222 | . . . . . . . 8 | |
12 | 10, 11 | syl 14 | . . . . . . 7 |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlembacc 6239 | . . . . . . . . . 10 |
14 | 13 | unissd 3760 | . . . . . . . . 9 |
15 | 5, 3 | tfr1onlemssrecs 6236 | . . . . . . . . 9 recs |
16 | 14, 15 | sstrd 3107 | . . . . . . . 8 recs |
17 | dmss 4738 | . . . . . . . 8 recs recs | |
18 | 16, 17 | syl 14 | . . . . . . 7 recs |
19 | 12, 18 | eqsstrrd 3134 | . . . . . 6 recs |
20 | 19 | sselda 3097 | . . . . 5 recs |
21 | eqid 2139 | . . . . . 6 | |
22 | 21 | tfrlem9 6216 | . . . . 5 recs recs recs |
23 | 20, 22 | syl 14 | . . . 4 recs recs |
24 | tfrfun 6217 | . . . . 5 recs | |
25 | 12 | eleq2d 2209 | . . . . . 6 |
26 | 25 | biimpar 295 | . . . . 5 |
27 | funssfv 5447 | . . . . 5 recs recs recs | |
28 | 24, 16, 26, 27 | mp3an2ani 1322 | . . . 4 recs |
29 | ordelon 4305 | . . . . . . . . . 10 | |
30 | 3, 8, 29 | syl2anc 408 | . . . . . . . . 9 |
31 | eloni 4297 | . . . . . . . . 9 | |
32 | 30, 31 | syl 14 | . . . . . . . 8 |
33 | ordelss 4301 | . . . . . . . 8 | |
34 | 32, 33 | sylan 281 | . . . . . . 7 |
35 | 12 | adantr 274 | . . . . . . 7 |
36 | 34, 35 | sseqtrrd 3136 | . . . . . 6 |
37 | fun2ssres 5166 | . . . . . 6 recs recs recs | |
38 | 24, 16, 36, 37 | mp3an2ani 1322 | . . . . 5 recs |
39 | 38 | fveq2d 5425 | . . . 4 recs |
40 | 23, 28, 39 | 3eqtr3d 2180 | . . 3 |
41 | 40 | ralrimiva 2505 | . 2 |
42 | fveq2 5421 | . . . 4 | |
43 | reseq2 4814 | . . . . 5 | |
44 | 43 | fveq2d 5425 | . . . 4 |
45 | 42, 44 | eqeq12d 2154 | . . 3 |
46 | 45 | cbvralv 2654 | . 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 2125 wral 2416 wrex 2417 cvv 2686 cun 3069 wss 3071 csn 3527 cop 3530 cuni 3736 word 4284 con0 4285 csuc 4287 cdm 4539 cres 4541 wfun 5117 wfn 5118 cfv 5123 recscrecs 6201 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-recs 6202 |
This theorem is referenced by: tfr1onlemex 6244 |
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