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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 6254. (Contributed by Jim Kingdon, 24-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f | recs |
tfrcl.g | |
tfrcl.x | |
tfrcl.ex | |
tfrcllemsucfn.1 | |
tfrcllemsucfn.3 | |
tfrcllemsucfn.4 | |
tfrcllemsucfn.5 |
Ref | Expression |
---|---|
tfrcllemsucfn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcllemsucfn.4 | . . 3 | |
2 | tfrcllemsucfn.3 | . . . 4 | |
3 | 2 | elexd 2694 | . . 3 |
4 | tfrcl.x | . . . . 5 | |
5 | ordelon 4300 | . . . . 5 | |
6 | 4, 2, 5 | syl2anc 408 | . . . 4 |
7 | eloni 4292 | . . . 4 | |
8 | ordirr 4452 | . . . 4 | |
9 | 6, 7, 8 | 3syl 17 | . . 3 |
10 | feq2 5251 | . . . . . . 7 | |
11 | 10 | imbi1d 230 | . . . . . 6 |
12 | 11 | albidv 1796 | . . . . 5 |
13 | tfrcl.ex | . . . . . . . 8 | |
14 | 13 | 3expia 1183 | . . . . . . 7 |
15 | 14 | alrimiv 1846 | . . . . . 6 |
16 | 15 | ralrimiva 2503 | . . . . 5 |
17 | 12, 16, 2 | rspcdva 2789 | . . . 4 |
18 | feq1 5250 | . . . . . 6 | |
19 | fveq2 5414 | . . . . . . 7 | |
20 | 19 | eleq1d 2206 | . . . . . 6 |
21 | 18, 20 | imbi12d 233 | . . . . 5 |
22 | 21 | spv 1832 | . . . 4 |
23 | 17, 1, 22 | sylc 62 | . . 3 |
24 | fsnunf 5613 | . . 3 | |
25 | 1, 3, 9, 23, 24 | syl121anc 1221 | . 2 |
26 | df-suc 4288 | . . 3 | |
27 | 26 | feq2i 5261 | . 2 |
28 | 25, 27 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 cvv 2681 cun 3064 csn 3522 cop 3525 word 4279 con0 4280 csuc 4282 cres 4536 wfun 5112 wf 5114 cfv 5118 recscrecs 6194 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: tfrcllemsucaccv 6244 tfrcllembfn 6247 |
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