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Theorem mapvalg 8832
Description: The value of set exponentiation. (𝐴m 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
mapvalg ((𝐴𝐶𝐵𝐷) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem mapvalg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapex 8828 . . 3 ((𝐵𝐷𝐴𝐶) → {𝑓𝑓:𝐵𝐴} ∈ V)
21ancoms 459 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐵𝐴} ∈ V)
3 elex 3492 . . 3 (𝐴𝐶𝐴 ∈ V)
4 elex 3492 . . 3 (𝐵𝐷𝐵 ∈ V)
5 feq3 6700 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑦𝑥𝑓:𝑦𝐴))
65abbidv 2801 . . . . 5 (𝑥 = 𝐴 → {𝑓𝑓:𝑦𝑥} = {𝑓𝑓:𝑦𝐴})
7 feq2 6699 . . . . . 6 (𝑦 = 𝐵 → (𝑓:𝑦𝐴𝑓:𝐵𝐴))
87abbidv 2801 . . . . 5 (𝑦 = 𝐵 → {𝑓𝑓:𝑦𝐴} = {𝑓𝑓:𝐵𝐴})
9 df-map 8824 . . . . 5 m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
106, 8, 9ovmpog 7569 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓𝑓:𝐵𝐴} ∈ V) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
11103expia 1121 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}))
123, 4, 11syl2an 596 . 2 ((𝐴𝐶𝐵𝐷) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}))
132, 12mpd 15 1 ((𝐴𝐶𝐵𝐷) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  Vcvv 3474  wf 6539  (class class class)co 7411  m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824
This theorem is referenced by:  mapval  8834  elmapg  8835  ixpconstg  8902  hashf1lem2  14421  efmndbasabf  18789  symgbasfi  19287  birthdaylem1  26680  birthdaylem2  26681  cnfex  44014
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