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Mirrors > Home > ILE Home > Th. List > tfrlemisucaccv | Unicode version |
Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfrlemisucfn.1 | |
tfrlemisucfn.2 | |
tfrlemisucfn.3 | |
tfrlemisucfn.4 | |
tfrlemisucfn.5 |
Ref | Expression |
---|---|
tfrlemisucaccv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlemisucfn.3 | . . . 4 | |
2 | suceloni 4417 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | tfrlemisucfn.1 | . . . 4 | |
5 | tfrlemisucfn.2 | . . . 4 | |
6 | tfrlemisucfn.4 | . . . 4 | |
7 | tfrlemisucfn.5 | . . . 4 | |
8 | 4, 5, 1, 6, 7 | tfrlemisucfn 6221 | . . 3 |
9 | vex 2689 | . . . . . 6 | |
10 | 9 | elsuc 4328 | . . . . 5 |
11 | vex 2689 | . . . . . . . . . . 11 | |
12 | 4, 11 | tfrlem3a 6207 | . . . . . . . . . 10 |
13 | 7, 12 | sylib 121 | . . . . . . . . 9 |
14 | simprrr 529 | . . . . . . . . . 10 | |
15 | simprrl 528 | . . . . . . . . . . . 12 | |
16 | 6 | adantr 274 | . . . . . . . . . . . 12 |
17 | fndmu 5224 | . . . . . . . . . . . 12 | |
18 | 15, 16, 17 | syl2anc 408 | . . . . . . . . . . 11 |
19 | 18 | raleqdv 2632 | . . . . . . . . . 10 |
20 | 14, 19 | mpbid 146 | . . . . . . . . 9 |
21 | 13, 20 | rexlimddv 2554 | . . . . . . . 8 |
22 | 21 | r19.21bi 2520 | . . . . . . 7 |
23 | elirrv 4463 | . . . . . . . . . . 11 | |
24 | elequ2 1691 | . . . . . . . . . . 11 | |
25 | 23, 24 | mtbiri 664 | . . . . . . . . . 10 |
26 | 25 | necon2ai 2362 | . . . . . . . . 9 |
27 | 26 | adantl 275 | . . . . . . . 8 |
28 | fvunsng 5614 | . . . . . . . 8 | |
29 | 9, 27, 28 | sylancr 410 | . . . . . . 7 |
30 | eloni 4297 | . . . . . . . . . . . 12 | |
31 | 1, 30 | syl 14 | . . . . . . . . . . 11 |
32 | ordelss 4301 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan 281 | . . . . . . . . . 10 |
34 | resabs1 4848 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 14 | . . . . . . . . 9 |
36 | elirrv 4463 | . . . . . . . . . . . 12 | |
37 | fsnunres 5622 | . . . . . . . . . . . 12 | |
38 | 6, 36, 37 | sylancl 409 | . . . . . . . . . . 11 |
39 | 38 | reseq1d 4818 | . . . . . . . . . 10 |
40 | 39 | adantr 274 | . . . . . . . . 9 |
41 | 35, 40 | eqtr3d 2174 | . . . . . . . 8 |
42 | 41 | fveq2d 5425 | . . . . . . 7 |
43 | 22, 29, 42 | 3eqtr4d 2182 | . . . . . 6 |
44 | 5 | tfrlem3-2d 6209 | . . . . . . . . . 10 |
45 | 44 | simprd 113 | . . . . . . . . 9 |
46 | fndm 5222 | . . . . . . . . . . . 12 | |
47 | 6, 46 | syl 14 | . . . . . . . . . . 11 |
48 | 47 | eleq2d 2209 | . . . . . . . . . 10 |
49 | 36, 48 | mtbiri 664 | . . . . . . . . 9 |
50 | fsnunfv 5621 | . . . . . . . . 9 | |
51 | 1, 45, 49, 50 | syl3anc 1216 | . . . . . . . 8 |
52 | 51 | adantr 274 | . . . . . . 7 |
53 | simpr 109 | . . . . . . . 8 | |
54 | 53 | fveq2d 5425 | . . . . . . 7 |
55 | reseq2 4814 | . . . . . . . . 9 | |
56 | 55, 38 | sylan9eqr 2194 | . . . . . . . 8 |
57 | 56 | fveq2d 5425 | . . . . . . 7 |
58 | 52, 54, 57 | 3eqtr4d 2182 | . . . . . 6 |
59 | 43, 58 | jaodan 786 | . . . . 5 |
60 | 10, 59 | sylan2b 285 | . . . 4 |
61 | 60 | ralrimiva 2505 | . . 3 |
62 | fneq2 5212 | . . . . 5 | |
63 | raleq 2626 | . . . . 5 | |
64 | 62, 63 | anbi12d 464 | . . . 4 |
65 | 64 | rspcev 2789 | . . 3 |
66 | 3, 8, 61, 65 | syl12anc 1214 | . 2 |
67 | vex 2689 | . . . . . 6 | |
68 | opexg 4150 | . . . . . 6 | |
69 | 67, 45, 68 | sylancr 410 | . . . . 5 |
70 | snexg 4108 | . . . . 5 | |
71 | 69, 70 | syl 14 | . . . 4 |
72 | unexg 4364 | . . . 4 | |
73 | 11, 71, 72 | sylancr 410 | . . 3 |
74 | 4 | tfrlem3ag 6206 | . . 3 |
75 | 73, 74 | syl 14 | . 2 |
76 | 66, 75 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wal 1329 wceq 1331 wcel 1480 cab 2125 wne 2308 wral 2416 wrex 2417 cvv 2686 cun 3069 wss 3071 csn 3527 cop 3530 word 4284 con0 4285 csuc 4287 cdm 4539 cres 4541 wfun 5117 wfn 5118 cfv 5123 |
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-v 2688 df-sbc 2910 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-br 3930 df-opab 3990 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-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: tfrlemibacc 6223 tfrlemi14d 6230 |
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