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 4171 | . . . 4 |
4 | 1, 3 | elmap 6655 | . . 3 |
5 | ffn 5347 | . . . . . . . 8 | |
6 | 2 | snid 3614 | . . . . . . . 8 |
7 | fneu 5302 | . . . . . . . 8 | |
8 | 5, 6, 7 | sylancl 411 | . . . . . . 7 |
9 | euabsn 3653 | . . . . . . . 8 | |
10 | imasng 4976 | . . . . . . . . . . . 12 | |
11 | 2, 10 | ax-mp 5 | . . . . . . . . . . 11 |
12 | fdm 5353 | . . . . . . . . . . . . 13 | |
13 | 12 | imaeq2d 4953 | . . . . . . . . . . . 12 |
14 | imadmrn 4963 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqtr3di 2218 | . . . . . . . . . . 11 |
16 | 11, 15 | eqtr3id 2217 | . . . . . . . . . 10 |
17 | 16 | eqeq1d 2179 | . . . . . . . . 9 |
18 | 17 | exbidv 1818 | . . . . . . . 8 |
19 | 9, 18 | syl5bb 191 | . . . . . . 7 |
20 | 8, 19 | mpbid 146 | . . . . . 6 |
21 | vex 2733 | . . . . . . . . . . 11 | |
22 | 21 | snid 3614 | . . . . . . . . . 10 |
23 | eleq2 2234 | . . . . . . . . . 10 | |
24 | 22, 23 | mpbiri 167 | . . . . . . . . 9 |
25 | frn 5356 | . . . . . . . . . 10 | |
26 | 25 | sseld 3146 | . . . . . . . . 9 |
27 | 24, 26 | syl5 32 | . . . . . . . 8 |
28 | dffn4 5426 | . . . . . . . . . . . 12 | |
29 | 5, 28 | sylib 121 | . . . . . . . . . . 11 |
30 | fof 5420 | . . . . . . . . . . 11 | |
31 | 29, 30 | syl 14 | . . . . . . . . . 10 |
32 | feq3 5332 | . . . . . . . . . 10 | |
33 | 31, 32 | syl5ibcom 154 | . . . . . . . . 9 |
34 | 2, 21 | fsn 5668 | . . . . . . . . 9 |
35 | 33, 34 | syl6ib 160 | . . . . . . . 8 |
36 | 27, 35 | jcad 305 | . . . . . . 7 |
37 | 36 | eximdv 1873 | . . . . . 6 |
38 | 20, 37 | mpd 13 | . . . . 5 |
39 | df-rex 2454 | . . . . 5 | |
40 | 38, 39 | sylibr 133 | . . . 4 |
41 | 2, 21 | f1osn 5482 | . . . . . . . . 9 |
42 | f1oeq1 5431 | . . . . . . . . 9 | |
43 | 41, 42 | mpbiri 167 | . . . . . . . 8 |
44 | f1of 5442 | . . . . . . . 8 | |
45 | 43, 44 | syl 14 | . . . . . . 7 |
46 | snssi 3724 | . . . . . . 7 | |
47 | fss 5359 | . . . . . . 7 | |
48 | 45, 46, 47 | syl2an 287 | . . . . . 6 |
49 | 48 | expcom 115 | . . . . 5 |
50 | 49 | rexlimiv 2581 | . . . 4 |
51 | 40, 50 | impbii 125 | . . 3 |
52 | 4, 51 | bitri 183 | . 2 |
53 | 52 | abbi2i 2285 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 weu 2019 wcel 2141 cab 2156 wrex 2449 cvv 2730 wss 3121 csn 3583 cop 3586 class class class wbr 3989 cdm 4611 crn 4612 cima 4614 wfn 5193 wf 5194 wfo 5196 wf1o 5197 (class class class)co 5853 cmap 6626 |
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-v 2732 df-sbc 2956 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-br 3990 df-opab 4051 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-map 6628 |
This theorem is referenced by: mapsnen 6789 |
Copyright terms: Public domain | W3C validator |