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Mirrors > Home > ILE Home > Th. List > mapdom1g | Unicode version |
Description: Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.) |
Ref | Expression |
---|---|
mapdom1g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 6639 | . . . . . 6 | |
2 | 1 | brrelex2i 4583 | . . . . 5 |
3 | domeng 6646 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 4 | ibi 175 | . . 3 |
6 | 5 | adantr 274 | . 2 |
7 | simpl 108 | . . . 4 | |
8 | enrefg 6658 | . . . . 5 | |
9 | 8 | adantl 275 | . . . 4 |
10 | mapen 6740 | . . . 4 | |
11 | 7, 9, 10 | syl2anr 288 | . . 3 |
12 | 2 | ad2antrr 479 | . . . . 5 |
13 | simprr 521 | . . . . 5 | |
14 | mapss 6585 | . . . . 5 | |
15 | 12, 13, 14 | syl2anc 408 | . . . 4 |
16 | fnmap 6549 | . . . . . . 7 | |
17 | elex 2697 | . . . . . . 7 | |
18 | fnovex 5804 | . . . . . . 7 | |
19 | 16, 2, 17, 18 | mp3an3an 1321 | . . . . . 6 |
20 | ssdomg 6672 | . . . . . 6 | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | 21 | adantr 274 | . . . 4 |
23 | 15, 22 | mpd 13 | . . 3 |
24 | endomtr 6684 | . . 3 | |
25 | 11, 23, 24 | syl2anc 408 | . 2 |
26 | 6, 25 | exlimddv 1870 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wex 1468 wcel 1480 cvv 2686 wss 3071 class class class wbr 3929 cxp 4537 wfn 5118 (class class class)co 5774 cmap 6542 cen 6632 cdom 6633 |
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-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 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-1st 6038 df-2nd 6039 df-map 6544 df-en 6635 df-dom 6636 |
This theorem is referenced by: (None) |
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