Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > freccllem | Unicode version |
Description: Lemma for freccl 6293. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.) |
Ref | Expression |
---|---|
freccl.a | |
freccl.cl | |
freccl.b | |
freccllem.g | recs |
Ref | Expression |
---|---|
freccllem | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-frec 6281 | . . . 4 frec recs | |
2 | freccllem.g | . . . . 5 recs | |
3 | 2 | reseq1i 4810 | . . . 4 recs |
4 | 1, 3 | eqtr4i 2161 | . . 3 frec |
5 | 4 | fveq1i 5415 | . 2 frec |
6 | freccl.b | . . . 4 | |
7 | fvres 5438 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | funmpt 5156 | . . . . 5 | |
10 | 9 | a1i 9 | . . . 4 |
11 | ordom 4515 | . . . . 5 | |
12 | 11 | a1i 9 | . . . 4 |
13 | vex 2684 | . . . . . 6 | |
14 | simp2 982 | . . . . . . 7 | |
15 | simp3 983 | . . . . . . 7 | |
16 | freccl.cl | . . . . . . . . 9 | |
17 | 16 | ralrimiva 2503 | . . . . . . . 8 |
18 | 17 | 3ad2ant1 1002 | . . . . . . 7 |
19 | freccl.a | . . . . . . . 8 | |
20 | 19 | 3ad2ant1 1002 | . . . . . . 7 |
21 | 14, 15, 18, 20 | frecabcl 6289 | . . . . . 6 |
22 | dmeq 4734 | . . . . . . . . . . . 12 | |
23 | 22 | eqeq1d 2146 | . . . . . . . . . . 11 |
24 | fveq1 5413 | . . . . . . . . . . . . 13 | |
25 | 24 | fveq2d 5418 | . . . . . . . . . . . 12 |
26 | 25 | eleq2d 2207 | . . . . . . . . . . 11 |
27 | 23, 26 | anbi12d 464 | . . . . . . . . . 10 |
28 | 27 | rexbidv 2436 | . . . . . . . . 9 |
29 | 22 | eqeq1d 2146 | . . . . . . . . . 10 |
30 | 29 | anbi1d 460 | . . . . . . . . 9 |
31 | 28, 30 | orbi12d 782 | . . . . . . . 8 |
32 | 31 | abbidv 2255 | . . . . . . 7 |
33 | eqid 2137 | . . . . . . 7 | |
34 | 32, 33 | fvmptg 5490 | . . . . . 6 |
35 | 13, 21, 34 | sylancr 410 | . . . . 5 |
36 | 35, 21 | eqeltrd 2214 | . . . 4 |
37 | limom 4522 | . . . . . . 7 | |
38 | limuni 4313 | . . . . . . 7 | |
39 | 37, 38 | ax-mp 5 | . . . . . 6 |
40 | 39 | eleq2i 2204 | . . . . 5 |
41 | peano2 4504 | . . . . . 6 | |
42 | 41 | adantl 275 | . . . . 5 |
43 | 40, 42 | sylan2br 286 | . . . 4 |
44 | 6, 39 | eleqtrdi 2230 | . . . 4 |
45 | 2, 10, 12, 36, 43, 44 | tfrcl 6254 | . . 3 |
46 | 8, 45 | eqeltrd 2214 | . 2 |
47 | 5, 46 | eqeltrid 2224 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 cab 2123 wral 2414 wrex 2415 cvv 2681 c0 3358 cuni 3731 cmpt 3984 word 4279 wlim 4281 csuc 4282 com 4499 cdm 4534 cres 4536 wfun 5112 wf 5114 cfv 5118 recscrecs 6194 freccfrec 6280 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-recs 6195 df-frec 6281 |
This theorem is referenced by: freccl 6293 |
Copyright terms: Public domain | W3C validator |