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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 6600 | . . 3 | |
2 | mapsnen.1 | . . 3 | |
3 | mapsnen.2 | . . . 4 | |
4 | 3 | snex 4146 | . . 3 |
5 | fnovex 5854 | . . 3 | |
6 | 1, 2, 4, 5 | mp3an 1319 | . 2 |
7 | vex 2715 | . . . 4 | |
8 | 7, 3 | fvex 5488 | . . 3 |
9 | 8 | a1i 9 | . 2 |
10 | vex 2715 | . . . . 5 | |
11 | 3, 10 | opex 4189 | . . . 4 |
12 | 11 | snex 4146 | . . 3 |
13 | 12 | a1i 9 | . 2 |
14 | 2, 3 | mapsn 6635 | . . . . . 6 |
15 | 14 | abeq2i 2268 | . . . . 5 |
16 | 15 | anbi1i 454 | . . . 4 |
17 | r19.41v 2613 | . . . 4 | |
18 | df-rex 2441 | . . . 4 | |
19 | 16, 17, 18 | 3bitr2i 207 | . . 3 |
20 | fveq1 5467 | . . . . . . . . . 10 | |
21 | vex 2715 | . . . . . . . . . . 11 | |
22 | 3, 21 | fvsn 5662 | . . . . . . . . . 10 |
23 | 20, 22 | eqtrdi 2206 | . . . . . . . . 9 |
24 | 23 | eqeq2d 2169 | . . . . . . . 8 |
25 | equcom 1686 | . . . . . . . 8 | |
26 | 24, 25 | bitrdi 195 | . . . . . . 7 |
27 | 26 | pm5.32i 450 | . . . . . 6 |
28 | 27 | anbi2i 453 | . . . . 5 |
29 | anass 399 | . . . . 5 | |
30 | ancom 264 | . . . . 5 | |
31 | 28, 29, 30 | 3bitr2i 207 | . . . 4 |
32 | 31 | exbii 1585 | . . 3 |
33 | eleq1w 2218 | . . . . 5 | |
34 | opeq2 3742 | . . . . . . 7 | |
35 | 34 | sneqd 3573 | . . . . . 6 |
36 | 35 | eqeq2d 2169 | . . . . 5 |
37 | 33, 36 | anbi12d 465 | . . . 4 |
38 | 10, 37 | ceqsexv 2751 | . . 3 |
39 | 19, 32, 38 | 3bitri 205 | . 2 |
40 | 6, 2, 9, 13, 39 | en2i 6715 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wex 1472 wcel 2128 wrex 2436 cvv 2712 csn 3560 cop 3563 class class class wbr 3965 cxp 4584 wfn 5165 cfv 5170 (class class class)co 5824 cmap 6593 cen 6683 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-map 6595 df-en 6686 |
This theorem is referenced by: (None) |
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