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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 5969. (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 4245 | . . . 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 5961 | . . 3 |
9 | vex 2604 | . . . . . 6 | |
10 | 9 | elsuc 4161 | . . . . 5 |
11 | vex 2604 | . . . . . . . . . . 11 | |
12 | 4, 11 | tfrlem3a 5948 | . . . . . . . . . 10 |
13 | 7, 12 | sylib 120 | . . . . . . . . 9 |
14 | simprrr 506 | . . . . . . . . . 10 | |
15 | simprrl 505 | . . . . . . . . . . . 12 | |
16 | 6 | adantr 270 | . . . . . . . . . . . 12 |
17 | fndmu 5020 | . . . . . . . . . . . 12 | |
18 | 15, 16, 17 | syl2anc 403 | . . . . . . . . . . 11 |
19 | 18 | raleqdv 2555 | . . . . . . . . . 10 |
20 | 14, 19 | mpbid 145 | . . . . . . . . 9 |
21 | 13, 20 | rexlimddv 2481 | . . . . . . . 8 |
22 | 21 | r19.21bi 2449 | . . . . . . 7 |
23 | elirrv 4291 | . . . . . . . . . . 11 | |
24 | elequ2 1641 | . . . . . . . . . . 11 | |
25 | 23, 24 | mtbiri 632 | . . . . . . . . . 10 |
26 | 25 | necon2ai 2299 | . . . . . . . . 9 |
27 | 26 | adantl 271 | . . . . . . . 8 |
28 | fvunsng 5378 | . . . . . . . 8 | |
29 | 9, 27, 28 | sylancr 405 | . . . . . . 7 |
30 | eloni 4130 | . . . . . . . . . . . 12 | |
31 | 1, 30 | syl 14 | . . . . . . . . . . 11 |
32 | ordelss 4134 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan 277 | . . . . . . . . . 10 |
34 | resabs1 4658 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 14 | . . . . . . . . 9 |
36 | elirrv 4291 | . . . . . . . . . . . 12 | |
37 | fsnunres 5385 | . . . . . . . . . . . 12 | |
38 | 6, 36, 37 | sylancl 404 | . . . . . . . . . . 11 |
39 | 38 | reseq1d 4629 | . . . . . . . . . 10 |
40 | 39 | adantr 270 | . . . . . . . . 9 |
41 | 35, 40 | eqtr3d 2115 | . . . . . . . 8 |
42 | 41 | fveq2d 5202 | . . . . . . 7 |
43 | 22, 29, 42 | 3eqtr4d 2123 | . . . . . 6 |
44 | 5 | tfrlem3-2d 5951 | . . . . . . . . . 10 |
45 | 44 | simprd 112 | . . . . . . . . 9 |
46 | fndm 5018 | . . . . . . . . . . . 12 | |
47 | 6, 46 | syl 14 | . . . . . . . . . . 11 |
48 | 47 | eleq2d 2148 | . . . . . . . . . 10 |
49 | 36, 48 | mtbiri 632 | . . . . . . . . 9 |
50 | fsnunfv 5384 | . . . . . . . . 9 | |
51 | 1, 45, 49, 50 | syl3anc 1169 | . . . . . . . 8 |
52 | 51 | adantr 270 | . . . . . . 7 |
53 | simpr 108 | . . . . . . . 8 | |
54 | 53 | fveq2d 5202 | . . . . . . 7 |
55 | reseq2 4625 | . . . . . . . . 9 | |
56 | 55, 38 | sylan9eqr 2135 | . . . . . . . 8 |
57 | 56 | fveq2d 5202 | . . . . . . 7 |
58 | 52, 54, 57 | 3eqtr4d 2123 | . . . . . 6 |
59 | 43, 58 | jaodan 743 | . . . . 5 |
60 | 10, 59 | sylan2b 281 | . . . 4 |
61 | 60 | ralrimiva 2434 | . . 3 |
62 | fneq2 5008 | . . . . 5 | |
63 | raleq 2549 | . . . . 5 | |
64 | 62, 63 | anbi12d 456 | . . . 4 |
65 | 64 | rspcev 2701 | . . 3 |
66 | 3, 8, 61, 65 | syl12anc 1167 | . 2 |
67 | vex 2604 | . . . . . 6 | |
68 | opexg 3983 | . . . . . 6 | |
69 | 67, 45, 68 | sylancr 405 | . . . . 5 |
70 | snexg 3956 | . . . . 5 | |
71 | 69, 70 | syl 14 | . . . 4 |
72 | unexg 4196 | . . . 4 | |
73 | 11, 71, 72 | sylancr 405 | . . 3 |
74 | 4 | tfrlem3ag 5947 | . . 3 |
75 | 73, 74 | syl 14 | . 2 |
76 | 66, 75 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wal 1282 wceq 1284 wcel 1433 cab 2067 wne 2245 wral 2348 wrex 2349 cvv 2601 cun 2971 wss 2973 csn 3398 cop 3401 word 4117 con0 4118 csuc 4120 cdm 4363 cres 4365 wfun 4916 wfn 4917 cfv 4922 |
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-v 2603 df-sbc 2816 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-br 3786 df-opab 3840 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-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: tfrlemibacc 5963 tfrlemi14d 5970 |
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