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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 6389. (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 2765 |
. . 3
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4 | tfrcl.x |
. . . . 5
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5 | ordelon 4401 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 2, 5 | syl2anc 411 |
. . . 4
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7 | eloni 4393 |
. . . 4
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8 | ordirr 4559 |
. . . 4
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9 | 6, 7, 8 | 3syl 17 |
. . 3
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10 | feq2 5368 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | imbi1d 231 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | albidv 1835 |
. . . . 5
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13 | tfrcl.ex |
. . . . . . . 8
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14 | 13 | 3expia 1207 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | alrimiv 1885 |
. . . . . 6
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16 | 15 | ralrimiva 2563 |
. . . . 5
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17 | 12, 16, 2 | rspcdva 2861 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | feq1 5367 |
. . . . . 6
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19 | fveq2 5534 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eleq1d 2258 |
. . . . . 6
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21 | 18, 20 | imbi12d 234 |
. . . . 5
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22 | 21 | spv 1871 |
. . . 4
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23 | 17, 1, 22 | sylc 62 |
. . 3
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24 | fsnunf 5737 |
. . 3
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25 | 1, 3, 9, 23, 24 | syl121anc 1254 |
. 2
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26 | df-suc 4389 |
. . 3
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27 | 26 | feq2i 5378 |
. 2
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28 | 25, 27 | sylibr 134 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 |
This theorem is referenced by: tfrcllemsucaccv 6379 tfrcllembfn 6382 |
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