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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 6269. (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 2702 |
. . 3
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4 | tfrcl.x |
. . . . 5
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5 | ordelon 4313 |
. . . . 5
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6 | 4, 2, 5 | syl2anc 409 |
. . . 4
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7 | eloni 4305 |
. . . 4
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8 | ordirr 4465 |
. . . 4
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9 | 6, 7, 8 | 3syl 17 |
. . 3
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10 | feq2 5264 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | imbi1d 230 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | albidv 1797 |
. . . . 5
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13 | tfrcl.ex |
. . . . . . . 8
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14 | 13 | 3expia 1184 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | alrimiv 1847 |
. . . . . 6
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16 | 15 | ralrimiva 2508 |
. . . . 5
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17 | 12, 16, 2 | rspcdva 2798 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | feq1 5263 |
. . . . . 6
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19 | fveq2 5429 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | eleq1d 2209 |
. . . . . 6
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21 | 18, 20 | imbi12d 233 |
. . . . 5
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22 | 21 | spv 1833 |
. . . 4
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23 | 17, 1, 22 | sylc 62 |
. . 3
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24 | fsnunf 5628 |
. . 3
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25 | 1, 3, 9, 23, 24 | syl121anc 1222 |
. 2
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26 | df-suc 4301 |
. . 3
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27 | 26 | feq2i 5274 |
. 2
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28 | 25, 27 | sylibr 133 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: tfrcllemsucaccv 6259 tfrcllembfn 6262 |
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