Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapvalg Unicode version

Theorem mapvalg 6777
 Description: The value of set exponentiation. is the set of all functions that map from to . Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
mapvalg
Distinct variable groups:   ,   ,
Dummy variables are mutually distinct and distinct from all other variables.
Allowed substitution hints:   ()   ()

Proof of Theorem mapvalg
StepHypRef Expression
1 mapex 6773 . . 3
21ancoms 441 . 2
3 elex 2797 . . 3
4 elex 2797 . . 3
5 feq3 5342 . . . . . 6
65abbidv 2398 . . . . 5
7 feq2 5341 . . . . . 6
87abbidv 2398 . . . . 5
9 df-map 6769 . . . . 5
106, 8, 9ovmpt2g 5943 . . . 4
11103expia 1155 . . 3
123, 4, 11syl2an 465 . 2
132, 12mpd 16 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1624   wcel 1685  cab 2270  cvv 2789  wf 5217  (class class class)co 5819   cmap 6767 This theorem is referenced by:  mapval  6779  elmapg  6780  ixpconstg  6820  hashf1lem2  11388  birthdaylem1  20240  birthdaylem2  20241  mapex2  24539  cnfex  27098 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-map 6769
 Copyright terms: Public domain W3C validator