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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 4147 | . . . 4 |
4 | 1, 3 | elmap 6623 | . . 3 |
5 | ffn 5320 | . . . . . . . 8 | |
6 | 2 | snid 3591 | . . . . . . . 8 |
7 | fneu 5275 | . . . . . . . 8 | |
8 | 5, 6, 7 | sylancl 410 | . . . . . . 7 |
9 | euabsn 3630 | . . . . . . . 8 | |
10 | imasng 4952 | . . . . . . . . . . . 12 | |
11 | 2, 10 | ax-mp 5 | . . . . . . . . . . 11 |
12 | imadmrn 4939 | . . . . . . . . . . . 12 | |
13 | fdm 5326 | . . . . . . . . . . . . 13 | |
14 | 13 | imaeq2d 4929 | . . . . . . . . . . . 12 |
15 | 12, 14 | syl5reqr 2205 | . . . . . . . . . . 11 |
16 | 11, 15 | eqtr3id 2204 | . . . . . . . . . 10 |
17 | 16 | eqeq1d 2166 | . . . . . . . . 9 |
18 | 17 | exbidv 1805 | . . . . . . . 8 |
19 | 9, 18 | syl5bb 191 | . . . . . . 7 |
20 | 8, 19 | mpbid 146 | . . . . . 6 |
21 | vex 2715 | . . . . . . . . . . 11 | |
22 | 21 | snid 3591 | . . . . . . . . . 10 |
23 | eleq2 2221 | . . . . . . . . . 10 | |
24 | 22, 23 | mpbiri 167 | . . . . . . . . 9 |
25 | frn 5329 | . . . . . . . . . 10 | |
26 | 25 | sseld 3127 | . . . . . . . . 9 |
27 | 24, 26 | syl5 32 | . . . . . . . 8 |
28 | dffn4 5399 | . . . . . . . . . . . 12 | |
29 | 5, 28 | sylib 121 | . . . . . . . . . . 11 |
30 | fof 5393 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | feq3 5305 | . . . . . . . . . 10 | |
33 | 31, 32 | syl5ibcom 154 | . . . . . . . . 9 |
34 | 2, 21 | fsn 5640 | . . . . . . . . 9 |
35 | 33, 34 | syl6ib 160 | . . . . . . . 8 |
36 | 27, 35 | jcad 305 | . . . . . . 7 |
37 | 36 | eximdv 1860 | . . . . . 6 |
38 | 20, 37 | mpd 13 | . . . . 5 |
39 | df-rex 2441 | . . . . 5 | |
40 | 38, 39 | sylibr 133 | . . . 4 |
41 | 2, 21 | f1osn 5455 | . . . . . . . . 9 |
42 | f1oeq1 5404 | . . . . . . . . 9 | |
43 | 41, 42 | mpbiri 167 | . . . . . . . 8 |
44 | f1of 5415 | . . . . . . . 8 | |
45 | 43, 44 | syl 14 | . . . . . . 7 |
46 | snssi 3701 | . . . . . . 7 | |
47 | fss 5332 | . . . . . . 7 | |
48 | 45, 46, 47 | syl2an 287 | . . . . . 6 |
49 | 48 | expcom 115 | . . . . 5 |
50 | 49 | rexlimiv 2568 | . . . 4 |
51 | 40, 50 | impbii 125 | . . 3 |
52 | 4, 51 | bitri 183 | . 2 |
53 | 52 | abbi2i 2272 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wex 1472 weu 2006 wcel 2128 cab 2143 wrex 2436 cvv 2712 wss 3102 csn 3560 cop 3563 class class class wbr 3966 cdm 4587 crn 4588 cima 4590 wfn 5166 wf 5167 wfo 5169 wf1o 5170 (class class class)co 5825 cmap 6594 |
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 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 |
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-v 2714 df-sbc 2938 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 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-map 6596 |
This theorem is referenced by: mapsnen 6757 |
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