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Mirrors > Home > ILE Home > Th. List > tfrcllemsucfn | Unicode version |
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6129. (Contributed by Jim Kingdon, 24-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f |
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tfrcl.g |
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tfrcl.x |
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tfrcl.ex |
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tfrcllemsucfn.1 |
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tfrcllemsucfn.3 |
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tfrcllemsucfn.4 |
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tfrcllemsucfn.5 |
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Ref | Expression |
---|---|
tfrcllemsucfn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcllemsucfn.4 |
. . 3
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2 | tfrcllemsucfn.3 |
. . . 4
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3 | 2 | elexd 2632 |
. . 3
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4 | tfrcl.x |
. . . . 5
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5 | ordelon 4210 |
. . . . 5
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6 | 4, 2, 5 | syl2anc 403 |
. . . 4
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7 | eloni 4202 |
. . . 4
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8 | ordirr 4358 |
. . . 4
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9 | 6, 7, 8 | 3syl 17 |
. . 3
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10 | feq2 5146 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | imbi1d 229 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | albidv 1752 |
. . . . 5
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13 | tfrcl.ex |
. . . . . . . 8
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14 | 13 | 3expia 1145 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | alrimiv 1802 |
. . . . . 6
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16 | 15 | ralrimiva 2446 |
. . . . 5
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17 | 12, 16, 2 | rspcdva 2727 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | feq1 5145 |
. . . . . 6
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19 | fveq2 5305 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eleq1d 2156 |
. . . . . 6
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21 | 18, 20 | imbi12d 232 |
. . . . 5
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22 | 21 | spv 1788 |
. . . 4
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23 | 17, 1, 22 | sylc 61 |
. . 3
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24 | fsnunf 5497 |
. . 3
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25 | 1, 3, 9, 23, 24 | syl121anc 1179 |
. 2
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26 | df-suc 4198 |
. . 3
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27 | 26 | feq2i 5155 |
. 2
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28 | 25, 27 | sylibr 132 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 |
This theorem is referenced by: tfrcllemsucaccv 6119 tfrcllembfn 6122 |
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