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Mirrors > Home > ILE Home > Th. List > mapsnen | Unicode version |
Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
mapsnen.1 | |
mapsnen.2 |
Ref | Expression |
---|---|
mapsnen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6549 | . . 3 | |
2 | mapsnen.1 | . . 3 | |
3 | mapsnen.2 | . . . 4 | |
4 | 3 | snex 4109 | . . 3 |
5 | fnovex 5804 | . . 3 | |
6 | 1, 2, 4, 5 | mp3an 1315 | . 2 |
7 | vex 2689 | . . . 4 | |
8 | 7, 3 | fvex 5441 | . . 3 |
9 | 8 | a1i 9 | . 2 |
10 | vex 2689 | . . . . 5 | |
11 | 3, 10 | opex 4151 | . . . 4 |
12 | 11 | snex 4109 | . . 3 |
13 | 12 | a1i 9 | . 2 |
14 | 2, 3 | mapsn 6584 | . . . . . 6 |
15 | 14 | abeq2i 2250 | . . . . 5 |
16 | 15 | anbi1i 453 | . . . 4 |
17 | r19.41v 2587 | . . . 4 | |
18 | df-rex 2422 | . . . 4 | |
19 | 16, 17, 18 | 3bitr2i 207 | . . 3 |
20 | fveq1 5420 | . . . . . . . . . 10 | |
21 | vex 2689 | . . . . . . . . . . 11 | |
22 | 3, 21 | fvsn 5615 | . . . . . . . . . 10 |
23 | 20, 22 | syl6eq 2188 | . . . . . . . . 9 |
24 | 23 | eqeq2d 2151 | . . . . . . . 8 |
25 | equcom 1682 | . . . . . . . 8 | |
26 | 24, 25 | syl6bb 195 | . . . . . . 7 |
27 | 26 | pm5.32i 449 | . . . . . 6 |
28 | 27 | anbi2i 452 | . . . . 5 |
29 | anass 398 | . . . . 5 | |
30 | ancom 264 | . . . . 5 | |
31 | 28, 29, 30 | 3bitr2i 207 | . . . 4 |
32 | 31 | exbii 1584 | . . 3 |
33 | eleq1w 2200 | . . . . 5 | |
34 | opeq2 3706 | . . . . . . 7 | |
35 | 34 | sneqd 3540 | . . . . . 6 |
36 | 35 | eqeq2d 2151 | . . . . 5 |
37 | 33, 36 | anbi12d 464 | . . . 4 |
38 | 10, 37 | ceqsexv 2725 | . . 3 |
39 | 19, 32, 38 | 3bitri 205 | . 2 |
40 | 6, 2, 9, 13, 39 | en2i 6664 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 wrex 2417 cvv 2686 csn 3527 cop 3530 class class class wbr 3929 cxp 4537 wfn 5118 cfv 5123 (class class class)co 5774 cmap 6542 cen 6632 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 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-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-en 6635 |
This theorem is referenced by: (None) |
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