Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6633 | . . 3 | |
2 | mapsnen.1 | . . 3 | |
3 | mapsnen.2 | . . . 4 | |
4 | 3 | snex 4171 | . . 3 |
5 | fnovex 5886 | . . 3 | |
6 | 1, 2, 4, 5 | mp3an 1332 | . 2 |
7 | vex 2733 | . . . 4 | |
8 | 7, 3 | fvex 5516 | . . 3 |
9 | 8 | a1i 9 | . 2 |
10 | vex 2733 | . . . . 5 | |
11 | 3, 10 | opex 4214 | . . . 4 |
12 | 11 | snex 4171 | . . 3 |
13 | 12 | a1i 9 | . 2 |
14 | 2, 3 | mapsn 6668 | . . . . . 6 |
15 | 14 | abeq2i 2281 | . . . . 5 |
16 | 15 | anbi1i 455 | . . . 4 |
17 | r19.41v 2626 | . . . 4 | |
18 | df-rex 2454 | . . . 4 | |
19 | 16, 17, 18 | 3bitr2i 207 | . . 3 |
20 | fveq1 5495 | . . . . . . . . . 10 | |
21 | vex 2733 | . . . . . . . . . . 11 | |
22 | 3, 21 | fvsn 5691 | . . . . . . . . . 10 |
23 | 20, 22 | eqtrdi 2219 | . . . . . . . . 9 |
24 | 23 | eqeq2d 2182 | . . . . . . . 8 |
25 | equcom 1699 | . . . . . . . 8 | |
26 | 24, 25 | bitrdi 195 | . . . . . . 7 |
27 | 26 | pm5.32i 451 | . . . . . 6 |
28 | 27 | anbi2i 454 | . . . . 5 |
29 | anass 399 | . . . . 5 | |
30 | ancom 264 | . . . . 5 | |
31 | 28, 29, 30 | 3bitr2i 207 | . . . 4 |
32 | 31 | exbii 1598 | . . 3 |
33 | eleq1w 2231 | . . . . 5 | |
34 | opeq2 3766 | . . . . . . 7 | |
35 | 34 | sneqd 3596 | . . . . . 6 |
36 | 35 | eqeq2d 2182 | . . . . 5 |
37 | 33, 36 | anbi12d 470 | . . . 4 |
38 | 10, 37 | ceqsexv 2769 | . . 3 |
39 | 19, 32, 38 | 3bitri 205 | . 2 |
40 | 6, 2, 9, 13, 39 | en2i 6748 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 wcel 2141 wrex 2449 cvv 2730 csn 3583 cop 3586 class class class wbr 3989 cxp 4609 wfn 5193 cfv 5198 (class class class)co 5853 cmap 6626 cen 6716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-en 6719 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |