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Mirrors > Home > ILE Home > Th. List > frecfcllem | Unicode version |
Description: Lemma for frecfcl 6270. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.) |
Ref | Expression |
---|---|
frecfcllem.g | recs |
Ref | Expression |
---|---|
frecfcllem | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecfcllem.g | . . . . . 6 recs | |
2 | funmpt 5131 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | ordom 4490 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | vex 2663 | . . . . . . . 8 | |
7 | simp2 967 | . . . . . . . . 9 | |
8 | simp3 968 | . . . . . . . . 9 | |
9 | simp1ll 1029 | . . . . . . . . . 10 | |
10 | fveq2 5389 | . . . . . . . . . . . 12 | |
11 | 10 | eleq1d 2186 | . . . . . . . . . . 11 |
12 | 11 | cbvralv 2631 | . . . . . . . . . 10 |
13 | 9, 12 | sylib 121 | . . . . . . . . 9 |
14 | simp1lr 1030 | . . . . . . . . 9 | |
15 | 7, 8, 13, 14 | frecabcl 6264 | . . . . . . . 8 |
16 | dmeq 4709 | . . . . . . . . . . . . . 14 | |
17 | 16 | eqeq1d 2126 | . . . . . . . . . . . . 13 |
18 | fveq1 5388 | . . . . . . . . . . . . . . 15 | |
19 | 18 | fveq2d 5393 | . . . . . . . . . . . . . 14 |
20 | 19 | eleq2d 2187 | . . . . . . . . . . . . 13 |
21 | 17, 20 | anbi12d 464 | . . . . . . . . . . . 12 |
22 | 21 | rexbidv 2415 | . . . . . . . . . . 11 |
23 | 16 | eqeq1d 2126 | . . . . . . . . . . . 12 |
24 | 23 | anbi1d 460 | . . . . . . . . . . 11 |
25 | 22, 24 | orbi12d 767 | . . . . . . . . . 10 |
26 | 25 | abbidv 2235 | . . . . . . . . 9 |
27 | eqid 2117 | . . . . . . . . 9 | |
28 | 26, 27 | fvmptg 5465 | . . . . . . . 8 |
29 | 6, 15, 28 | sylancr 410 | . . . . . . 7 |
30 | 29, 15 | eqeltrd 2194 | . . . . . 6 |
31 | limom 4497 | . . . . . . . . . 10 | |
32 | limuni 4288 | . . . . . . . . . 10 | |
33 | 31, 32 | ax-mp 5 | . . . . . . . . 9 |
34 | 33 | eleq2i 2184 | . . . . . . . 8 |
35 | peano2 4479 | . . . . . . . 8 | |
36 | 34, 35 | sylbir 134 | . . . . . . 7 |
37 | 36 | adantl 275 | . . . . . 6 |
38 | 33 | eleq2i 2184 | . . . . . . . 8 |
39 | 38 | biimpi 119 | . . . . . . 7 |
40 | 39 | adantl 275 | . . . . . 6 |
41 | 1, 3, 5, 30, 37, 40 | tfrcldm 6228 | . . . . 5 |
42 | 1, 3, 5, 30, 37, 40 | tfrcl 6229 | . . . . 5 |
43 | 41, 42 | jca 304 | . . . 4 |
44 | 43 | ralrimiva 2482 | . . 3 |
45 | tfrfun 6185 | . . . . 5 recs | |
46 | 1 | funeqi 5114 | . . . . 5 recs |
47 | 45, 46 | mpbir 145 | . . . 4 |
48 | ffvresb 5551 | . . . 4 | |
49 | 47, 48 | ax-mp 5 | . . 3 |
50 | 44, 49 | sylibr 133 | . 2 |
51 | df-frec 6256 | . . . 4 frec recs | |
52 | 1 | reseq1i 4785 | . . . 4 recs |
53 | 51, 52 | eqtr4i 2141 | . . 3 frec |
54 | 53 | feq1i 5235 | . 2 frec |
55 | 50, 54 | sylibr 133 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 w3a 947 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 cvv 2660 c0 3333 cuni 3706 cmpt 3959 word 4254 wlim 4256 csuc 4257 com 4474 cdm 4509 cres 4511 wfun 5087 wf 5089 cfv 5093 recscrecs 6169 freccfrec 6255 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-recs 6170 df-frec 6256 |
This theorem is referenced by: frecfcl 6270 |
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