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Theorem mapvalg 8782
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 8778 . . 3 ((𝐵𝐷𝐴𝐶) → {𝑓𝑓:𝐵𝐴} ∈ V)
21ancoms 460 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐵𝐴} ∈ V)
3 elex 3466 . . 3 (𝐴𝐶𝐴 ∈ V)
4 elex 3466 . . 3 (𝐵𝐷𝐵 ∈ V)
5 feq3 6656 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑦𝑥𝑓:𝑦𝐴))
65abbidv 2806 . . . . 5 (𝑥 = 𝐴 → {𝑓𝑓:𝑦𝑥} = {𝑓𝑓:𝑦𝐴})
7 feq2 6655 . . . . . 6 (𝑦 = 𝐵 → (𝑓:𝑦𝐴𝑓:𝐵𝐴))
87abbidv 2806 . . . . 5 (𝑦 = 𝐵 → {𝑓𝑓:𝑦𝐴} = {𝑓𝑓:𝐵𝐴})
9 df-map 8774 . . . . 5 m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
106, 8, 9ovmpog 7519 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓𝑓:𝐵𝐴} ∈ V) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
11103expia 1122 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}))
123, 4, 11syl2an 597 . 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 397   = wceq 1542  wcel 2107  {cab 2714  Vcvv 3448  wf 6497  (class class class)co 7362  m cmap 8772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774
This theorem is referenced by:  mapval  8784  elmapg  8785  ixpconstg  8851  hashf1lem2  14362  efmndbasabf  18689  symgbasfi  19167  birthdaylem1  26317  birthdaylem2  26318  cnfex  43307
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