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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 6745 |
. . 3
| |
| 2 | 1 | eleq2d 2275 |
. 2
|
| 3 | fex2 5444 |
. . . . 5
| |
| 4 | 3 | 3com13 1211 |
. . . 4
|
| 5 | 4 | 3expia 1208 |
. . 3
|
| 6 | feq1 5408 |
. . . 4
| |
| 7 | 6 | elab3g 2924 |
. . 3
|
| 8 | 5, 7 | syl 14 |
. 2
|
| 9 | 2, 8 | bitrd 188 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-ral 2489 df-rex 2490 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-map 6737 |
| This theorem is referenced by: elmapd 6749 mapdm0 6750 elmapi 6757 elmap 6764 map0e 6773 map0g 6775 fdiagfn 6779 ixpssmap2g 6814 map1 6904 mapxpen 6945 infnninf 7226 isomnimap 7239 enomnilem 7240 ismkvmap 7256 enmkvlem 7263 iswomnimap 7268 enwomnilem 7271 hashfacen 10981 wrdnval 11024 omctfn 12814 pwselbasb 13125 psrbag 14431 iscn 14669 iscnp 14671 cndis 14713 ispsmet 14795 ismet 14816 isxmet 14817 elcncf 15045 elply2 15207 plyf 15209 elplyr 15212 plyaddlem 15221 plymullem 15222 plyco 15231 nnsf 15942 |
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