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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 5946. (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 4227 | . . . 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 5938 | . . 3 |
9 | vex 2560 | . . . . . 6 | |
10 | 9 | elsuc 4143 | . . . . 5 |
11 | vex 2560 | . . . . . . . . . . 11 | |
12 | 4, 11 | tfrlem3a 5925 | . . . . . . . . . 10 |
13 | 7, 12 | sylib 127 | . . . . . . . . 9 |
14 | simprrr 492 | . . . . . . . . . 10 | |
15 | simprrl 491 | . . . . . . . . . . . 12 | |
16 | 6 | adantr 261 | . . . . . . . . . . . 12 |
17 | fndmu 5000 | . . . . . . . . . . . 12 | |
18 | 15, 16, 17 | syl2anc 391 | . . . . . . . . . . 11 |
19 | 18 | raleqdv 2511 | . . . . . . . . . 10 |
20 | 14, 19 | mpbid 135 | . . . . . . . . 9 |
21 | 13, 20 | rexlimddv 2437 | . . . . . . . 8 |
22 | 21 | r19.21bi 2407 | . . . . . . 7 |
23 | elirrv 4272 | . . . . . . . . . . 11 | |
24 | elequ2 1601 | . . . . . . . . . . 11 | |
25 | 23, 24 | mtbiri 600 | . . . . . . . . . 10 |
26 | 25 | necon2ai 2259 | . . . . . . . . 9 |
27 | 26 | adantl 262 | . . . . . . . 8 |
28 | fvunsng 5357 | . . . . . . . 8 | |
29 | 9, 27, 28 | sylancr 393 | . . . . . . 7 |
30 | eloni 4112 | . . . . . . . . . . . 12 | |
31 | 1, 30 | syl 14 | . . . . . . . . . . 11 |
32 | ordelss 4116 | . . . . . . . . . . 11 | |
33 | 31, 32 | sylan 267 | . . . . . . . . . 10 |
34 | resabs1 4640 | . . . . . . . . . 10 | |
35 | 33, 34 | syl 14 | . . . . . . . . 9 |
36 | elirrv 4272 | . . . . . . . . . . . 12 | |
37 | fsnunres 5364 | . . . . . . . . . . . 12 | |
38 | 6, 36, 37 | sylancl 392 | . . . . . . . . . . 11 |
39 | 38 | reseq1d 4611 | . . . . . . . . . 10 |
40 | 39 | adantr 261 | . . . . . . . . 9 |
41 | 35, 40 | eqtr3d 2074 | . . . . . . . 8 |
42 | 41 | fveq2d 5182 | . . . . . . 7 |
43 | 22, 29, 42 | 3eqtr4d 2082 | . . . . . 6 |
44 | 5 | tfrlem3-2d 5928 | . . . . . . . . . 10 |
45 | 44 | simprd 107 | . . . . . . . . 9 |
46 | fndm 4998 | . . . . . . . . . . . 12 | |
47 | 6, 46 | syl 14 | . . . . . . . . . . 11 |
48 | 47 | eleq2d 2107 | . . . . . . . . . 10 |
49 | 36, 48 | mtbiri 600 | . . . . . . . . 9 |
50 | fsnunfv 5363 | . . . . . . . . 9 | |
51 | 1, 45, 49, 50 | syl3anc 1135 | . . . . . . . 8 |
52 | 51 | adantr 261 | . . . . . . 7 |
53 | simpr 103 | . . . . . . . 8 | |
54 | 53 | fveq2d 5182 | . . . . . . 7 |
55 | reseq2 4607 | . . . . . . . . 9 | |
56 | 55, 38 | sylan9eqr 2094 | . . . . . . . 8 |
57 | 56 | fveq2d 5182 | . . . . . . 7 |
58 | 52, 54, 57 | 3eqtr4d 2082 | . . . . . 6 |
59 | 43, 58 | jaodan 710 | . . . . 5 |
60 | 10, 59 | sylan2b 271 | . . . 4 |
61 | 60 | ralrimiva 2392 | . . 3 |
62 | fneq2 4988 | . . . . 5 | |
63 | raleq 2505 | . . . . 5 | |
64 | 62, 63 | anbi12d 442 | . . . 4 |
65 | 64 | rspcev 2656 | . . 3 |
66 | 3, 8, 61, 65 | syl12anc 1133 | . 2 |
67 | vex 2560 | . . . . . 6 | |
68 | opexg 3964 | . . . . . 6 | |
69 | 67, 45, 68 | sylancr 393 | . . . . 5 |
70 | snexg 3936 | . . . . 5 | |
71 | 69, 70 | syl 14 | . . . 4 |
72 | unexg 4178 | . . . 4 | |
73 | 11, 71, 72 | sylancr 393 | . . 3 |
74 | 4 | tfrlem3ag 5924 | . . 3 |
75 | 73, 74 | syl 14 | . 2 |
76 | 66, 75 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 wal 1241 wceq 1243 wcel 1393 cab 2026 wne 2204 wral 2306 wrex 2307 cvv 2557 cun 2915 wss 2917 csn 3375 cop 3378 word 4099 con0 4100 csuc 4102 cdm 4345 cres 4347 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: tfrlemibacc 5940 tfrlemi14d 5947 |
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