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Mirrors > Home > ILE Home > Th. List > map1 | Unicode version |
Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) |
Ref | Expression |
---|---|
map1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6631 | . . 3 | |
2 | 1oex 6401 | . . 3 | |
3 | elex 2741 | . . 3 | |
4 | fnovex 5884 | . . 3 | |
5 | 1, 2, 3, 4 | mp3an12i 1336 | . 2 |
6 | 2 | a1i 9 | . 2 |
7 | 0ex 4114 | . . 3 | |
8 | 7 | 2a1i 27 | . 2 |
9 | p0ex 4172 | . . . 4 | |
10 | xpexg 4723 | . . . 4 | |
11 | 9, 10 | mpan2 423 | . . 3 |
12 | 11 | a1d 22 | . 2 |
13 | el1o 6414 | . . . . 5 | |
14 | 13 | a1i 9 | . . . 4 |
15 | df1o2 6406 | . . . . . . . 8 | |
16 | 15 | oveq1i 5861 | . . . . . . 7 |
17 | 16 | eleq2i 2237 | . . . . . 6 |
18 | elmapg 6637 | . . . . . . 7 | |
19 | 9, 18 | mpan 422 | . . . . . 6 |
20 | 17, 19 | syl5bb 191 | . . . . 5 |
21 | 7 | fconst2 5711 | . . . . 5 |
22 | 20, 21 | bitr2di 196 | . . . 4 |
23 | 14, 22 | anbi12d 470 | . . 3 |
24 | ancom 264 | . . 3 | |
25 | 23, 24 | bitr2di 196 | . 2 |
26 | 5, 6, 8, 12, 25 | en2d 6744 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cvv 2730 c0 3414 csn 3581 class class class wbr 3987 cxp 4607 wfn 5191 wf 5192 (class class class)co 5851 c1o 6386 cmap 6624 cen 6714 |
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 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
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-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-1o 6393 df-map 6626 df-en 6717 |
This theorem is referenced by: (None) |
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