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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 6632 | . . 3 | |
2 | 1 | eleq2d 2240 | . 2 |
3 | fex2 5364 | . . . . 5 | |
4 | 3 | 3com13 1203 | . . . 4 |
5 | 4 | 3expia 1200 | . . 3 |
6 | feq1 5328 | . . . 4 | |
7 | 6 | elab3g 2881 | . . 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 2141 cab 2156 cvv 2730 wf 5192 (class class class)co 5850 cmap 6622 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
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-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-map 6624 |
This theorem is referenced by: elmapd 6636 mapdm0 6637 elmapi 6644 elmap 6651 map0e 6660 map0g 6662 fdiagfn 6666 ixpssmap2g 6701 map1 6786 mapxpen 6822 infnninf 7096 isomnimap 7109 enomnilem 7110 ismkvmap 7126 enmkvlem 7133 iswomnimap 7138 enwomnilem 7141 hashfacen 10758 omctfn 12385 iscn 12950 iscnp 12952 cndis 12994 ispsmet 13076 ismet 13097 isxmet 13098 elcncf 13313 nnsf 13998 |
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