Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > frecfcllem | Unicode version |
Description: Lemma for frecfcl 6302. 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 5161 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | ordom 4520 | . . . . . . 7 | |
5 | 4 | a1i 9 | . . . . . 6 |
6 | vex 2689 | . . . . . . . 8 | |
7 | simp2 982 | . . . . . . . . 9 | |
8 | simp3 983 | . . . . . . . . 9 | |
9 | simp1ll 1044 | . . . . . . . . . 10 | |
10 | fveq2 5421 | . . . . . . . . . . . 12 | |
11 | 10 | eleq1d 2208 | . . . . . . . . . . 11 |
12 | 11 | cbvralv 2654 | . . . . . . . . . 10 |
13 | 9, 12 | sylib 121 | . . . . . . . . 9 |
14 | simp1lr 1045 | . . . . . . . . 9 | |
15 | 7, 8, 13, 14 | frecabcl 6296 | . . . . . . . 8 |
16 | dmeq 4739 | . . . . . . . . . . . . . 14 | |
17 | 16 | eqeq1d 2148 | . . . . . . . . . . . . 13 |
18 | fveq1 5420 | . . . . . . . . . . . . . . 15 | |
19 | 18 | fveq2d 5425 | . . . . . . . . . . . . . 14 |
20 | 19 | eleq2d 2209 | . . . . . . . . . . . . 13 |
21 | 17, 20 | anbi12d 464 | . . . . . . . . . . . 12 |
22 | 21 | rexbidv 2438 | . . . . . . . . . . 11 |
23 | 16 | eqeq1d 2148 | . . . . . . . . . . . 12 |
24 | 23 | anbi1d 460 | . . . . . . . . . . 11 |
25 | 22, 24 | orbi12d 782 | . . . . . . . . . 10 |
26 | 25 | abbidv 2257 | . . . . . . . . 9 |
27 | eqid 2139 | . . . . . . . . 9 | |
28 | 26, 27 | fvmptg 5497 | . . . . . . . 8 |
29 | 6, 15, 28 | sylancr 410 | . . . . . . 7 |
30 | 29, 15 | eqeltrd 2216 | . . . . . 6 |
31 | limom 4527 | . . . . . . . . . 10 | |
32 | limuni 4318 | . . . . . . . . . 10 | |
33 | 31, 32 | ax-mp 5 | . . . . . . . . 9 |
34 | 33 | eleq2i 2206 | . . . . . . . 8 |
35 | peano2 4509 | . . . . . . . 8 | |
36 | 34, 35 | sylbir 134 | . . . . . . 7 |
37 | 36 | adantl 275 | . . . . . 6 |
38 | 33 | eleq2i 2206 | . . . . . . . 8 |
39 | 38 | biimpi 119 | . . . . . . 7 |
40 | 39 | adantl 275 | . . . . . 6 |
41 | 1, 3, 5, 30, 37, 40 | tfrcldm 6260 | . . . . 5 |
42 | 1, 3, 5, 30, 37, 40 | tfrcl 6261 | . . . . 5 |
43 | 41, 42 | jca 304 | . . . 4 |
44 | 43 | ralrimiva 2505 | . . 3 |
45 | tfrfun 6217 | . . . . 5 recs | |
46 | 1 | funeqi 5144 | . . . . 5 recs |
47 | 45, 46 | mpbir 145 | . . . 4 |
48 | ffvresb 5583 | . . . 4 | |
49 | 47, 48 | ax-mp 5 | . . 3 |
50 | 44, 49 | sylibr 133 | . 2 |
51 | df-frec 6288 | . . . 4 frec recs | |
52 | 1 | reseq1i 4815 | . . . 4 recs |
53 | 51, 52 | eqtr4i 2163 | . . 3 frec |
54 | 53 | feq1i 5265 | . 2 frec |
55 | 50, 54 | sylibr 133 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 c0 3363 cuni 3736 cmpt 3989 word 4284 wlim 4286 csuc 4287 com 4504 cdm 4539 cres 4541 wfun 5117 wf 5119 cfv 5123 recscrecs 6201 freccfrec 6287 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-recs 6202 df-frec 6288 |
This theorem is referenced by: frecfcl 6302 |
Copyright terms: Public domain | W3C validator |