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Mirrors > Home > ILE Home > Th. List > elmapg | Unicode version |
Description: Membership relation for set exponentiation. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
elmapg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapvalg 6520 | . . 3 | |
2 | 1 | eleq2d 2187 | . 2 |
3 | fex2 5261 | . . . . 5 | |
4 | 3 | 3com13 1171 | . . . 4 |
5 | 4 | 3expia 1168 | . . 3 |
6 | feq1 5225 | . . . 4 | |
7 | 6 | elab3g 2808 | . . 3 |
8 | 5, 7 | syl 14 | . 2 |
9 | 2, 8 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 cab 2103 cvv 2660 wf 5089 (class class class)co 5742 cmap 6510 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-map 6512 |
This theorem is referenced by: elmapd 6524 mapdm0 6525 elmapi 6532 elmap 6539 map0e 6548 map0g 6550 fdiagfn 6554 ixpssmap2g 6589 map1 6674 mapxpen 6710 isomnimap 6977 enomnilem 6978 ismkvmap 6996 hashfacen 10547 iscn 12293 iscnp 12295 cndis 12337 ispsmet 12419 ismet 12440 isxmet 12441 elcncf 12656 nnsf 13126 |
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