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Mirrors > Home > ILE Home > Th. List > tfrlemiubacc | Unicode version |
Description: The union of satisfies the recursion rule (lemma for tfrlemi1 5969). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfrlemisucfn.1 | |
tfrlemisucfn.2 | |
tfrlemi1.3 | |
tfrlemi1.4 | |
tfrlemi1.5 |
Ref | Expression |
---|---|
tfrlemiubacc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlemisucfn.1 | . . . . . . . . 9 | |
2 | tfrlemisucfn.2 | . . . . . . . . 9 | |
3 | tfrlemi1.3 | . . . . . . . . 9 | |
4 | tfrlemi1.4 | . . . . . . . . 9 | |
5 | tfrlemi1.5 | . . . . . . . . 9 | |
6 | 1, 2, 3, 4, 5 | tfrlemibfn 5965 | . . . . . . . 8 |
7 | fndm 5018 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | 1, 2, 3, 4, 5 | tfrlemibacc 5963 | . . . . . . . . . 10 |
10 | 9 | unissd 3625 | . . . . . . . . 9 |
11 | 1 | recsfval 5954 | . . . . . . . . 9 recs |
12 | 10, 11 | syl6sseqr 3046 | . . . . . . . 8 recs |
13 | dmss 4552 | . . . . . . . 8 recs recs | |
14 | 12, 13 | syl 14 | . . . . . . 7 recs |
15 | 8, 14 | eqsstr3d 3034 | . . . . . 6 recs |
16 | 15 | sselda 2999 | . . . . 5 recs |
17 | 1 | tfrlem9 5958 | . . . . 5 recs recs recs |
18 | 16, 17 | syl 14 | . . . 4 recs recs |
19 | 1 | tfrlem7 5956 | . . . . . 6 recs |
20 | 19 | a1i 9 | . . . . 5 recs |
21 | 12 | adantr 270 | . . . . 5 recs |
22 | 8 | eleq2d 2148 | . . . . . 6 |
23 | 22 | biimpar 291 | . . . . 5 |
24 | funssfv 5220 | . . . . 5 recs recs recs | |
25 | 20, 21, 23, 24 | syl3anc 1169 | . . . 4 recs |
26 | eloni 4130 | . . . . . . . . 9 | |
27 | 4, 26 | syl 14 | . . . . . . . 8 |
28 | ordelss 4134 | . . . . . . . 8 | |
29 | 27, 28 | sylan 277 | . . . . . . 7 |
30 | 8 | adantr 270 | . . . . . . 7 |
31 | 29, 30 | sseqtr4d 3036 | . . . . . 6 |
32 | fun2ssres 4963 | . . . . . 6 recs recs recs | |
33 | 20, 21, 31, 32 | syl3anc 1169 | . . . . 5 recs |
34 | 33 | fveq2d 5202 | . . . 4 recs |
35 | 18, 25, 34 | 3eqtr3d 2121 | . . 3 |
36 | 35 | ralrimiva 2434 | . 2 |
37 | fveq2 5198 | . . . 4 | |
38 | reseq2 4625 | . . . . 5 | |
39 | 38 | fveq2d 5202 | . . . 4 |
40 | 37, 39 | eqeq12d 2095 | . . 3 |
41 | 40 | cbvralv 2577 | . 2 |
42 | 36, 41 | sylibr 132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wal 1282 wceq 1284 wex 1421 wcel 1433 cab 2067 wral 2348 wrex 2349 cvv 2601 cun 2971 wss 2973 csn 3398 cop 3401 cuni 3601 word 4117 con0 4118 cdm 4363 cres 4365 wfun 4916 wfn 4917 cfv 4922 recscrecs 5942 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-recs 5943 |
This theorem is referenced by: tfrlemiex 5968 |
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