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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 6636 | . . 3 | |
2 | 1 | eleq2d 2240 | . 2 |
3 | fex2 5366 | . . . . 5 | |
4 | 3 | 3com13 1203 | . . . 4 |
5 | 4 | 3expia 1200 | . . 3 |
6 | feq1 5330 | . . . 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 5194 (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-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-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628 |
This theorem is referenced by: elmapd 6640 mapdm0 6641 elmapi 6648 elmap 6655 map0e 6664 map0g 6666 fdiagfn 6670 ixpssmap2g 6705 map1 6790 mapxpen 6826 infnninf 7100 isomnimap 7113 enomnilem 7114 ismkvmap 7130 enmkvlem 7137 iswomnimap 7142 enwomnilem 7145 hashfacen 10771 omctfn 12398 iscn 12991 iscnp 12993 cndis 13035 ispsmet 13117 ismet 13138 isxmet 13139 elcncf 13354 nnsf 14038 |
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