Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mapsn | Unicode version |
Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
map0.1 | |
map0.2 |
Ref | Expression |
---|---|
mapsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | map0.1 | . . . 4 | |
2 | map0.2 | . . . . 5 | |
3 | 2 | snex 4109 | . . . 4 |
4 | 1, 3 | elmap 6571 | . . 3 |
5 | ffn 5272 | . . . . . . . 8 | |
6 | 2 | snid 3556 | . . . . . . . 8 |
7 | fneu 5227 | . . . . . . . 8 | |
8 | 5, 6, 7 | sylancl 409 | . . . . . . 7 |
9 | euabsn 3593 | . . . . . . . 8 | |
10 | imasng 4904 | . . . . . . . . . . . 12 | |
11 | 2, 10 | ax-mp 5 | . . . . . . . . . . 11 |
12 | imadmrn 4891 | . . . . . . . . . . . 12 | |
13 | fdm 5278 | . . . . . . . . . . . . 13 | |
14 | 13 | imaeq2d 4881 | . . . . . . . . . . . 12 |
15 | 12, 14 | syl5reqr 2187 | . . . . . . . . . . 11 |
16 | 11, 15 | syl5eqr 2186 | . . . . . . . . . 10 |
17 | 16 | eqeq1d 2148 | . . . . . . . . 9 |
18 | 17 | exbidv 1797 | . . . . . . . 8 |
19 | 9, 18 | syl5bb 191 | . . . . . . 7 |
20 | 8, 19 | mpbid 146 | . . . . . 6 |
21 | vex 2689 | . . . . . . . . . . 11 | |
22 | 21 | snid 3556 | . . . . . . . . . 10 |
23 | eleq2 2203 | . . . . . . . . . 10 | |
24 | 22, 23 | mpbiri 167 | . . . . . . . . 9 |
25 | frn 5281 | . . . . . . . . . 10 | |
26 | 25 | sseld 3096 | . . . . . . . . 9 |
27 | 24, 26 | syl5 32 | . . . . . . . 8 |
28 | dffn4 5351 | . . . . . . . . . . . 12 | |
29 | 5, 28 | sylib 121 | . . . . . . . . . . 11 |
30 | fof 5345 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | feq3 5257 | . . . . . . . . . 10 | |
33 | 31, 32 | syl5ibcom 154 | . . . . . . . . 9 |
34 | 2, 21 | fsn 5592 | . . . . . . . . 9 |
35 | 33, 34 | syl6ib 160 | . . . . . . . 8 |
36 | 27, 35 | jcad 305 | . . . . . . 7 |
37 | 36 | eximdv 1852 | . . . . . 6 |
38 | 20, 37 | mpd 13 | . . . . 5 |
39 | df-rex 2422 | . . . . 5 | |
40 | 38, 39 | sylibr 133 | . . . 4 |
41 | 2, 21 | f1osn 5407 | . . . . . . . . 9 |
42 | f1oeq1 5356 | . . . . . . . . 9 | |
43 | 41, 42 | mpbiri 167 | . . . . . . . 8 |
44 | f1of 5367 | . . . . . . . 8 | |
45 | 43, 44 | syl 14 | . . . . . . 7 |
46 | snssi 3664 | . . . . . . 7 | |
47 | fss 5284 | . . . . . . 7 | |
48 | 45, 46, 47 | syl2an 287 | . . . . . 6 |
49 | 48 | expcom 115 | . . . . 5 |
50 | 49 | rexlimiv 2543 | . . . 4 |
51 | 40, 50 | impbii 125 | . . 3 |
52 | 4, 51 | bitri 183 | . 2 |
53 | 52 | abbi2i 2254 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wex 1468 wcel 1480 weu 1999 cab 2125 wrex 2417 cvv 2686 wss 3071 csn 3527 cop 3530 class class class wbr 3929 cdm 4539 crn 4540 cima 4542 wfn 5118 wf 5119 wfo 5121 wf1o 5122 (class class class)co 5774 cmap 6542 |
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-v 2688 df-sbc 2910 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-br 3930 df-opab 3990 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-map 6544 |
This theorem is referenced by: mapsnen 6705 |
Copyright terms: Public domain | W3C validator |