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Theorem mapvalg 6520
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 6516 . . 3 ((𝐵𝐷𝐴𝐶) → {𝑓𝑓:𝐵𝐴} ∈ V)
21ancoms 266 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐵𝐴} ∈ V)
3 elex 2671 . . 3 (𝐴𝐶𝐴 ∈ V)
4 elex 2671 . . 3 (𝐵𝐷𝐵 ∈ V)
5 feq3 5227 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑦𝑥𝑓:𝑦𝐴))
65abbidv 2235 . . . . 5 (𝑥 = 𝐴 → {𝑓𝑓:𝑦𝑥} = {𝑓𝑓:𝑦𝐴})
7 feq2 5226 . . . . . 6 (𝑦 = 𝐵 → (𝑓:𝑦𝐴𝑓:𝐵𝐴))
87abbidv 2235 . . . . 5 (𝑦 = 𝐵 → {𝑓𝑓:𝑦𝐴} = {𝑓𝑓:𝐵𝐴})
9 df-map 6512 . . . . 5 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
106, 8, 9ovmpog 5873 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓𝑓:𝐵𝐴} ∈ V) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
11103expia 1168 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}))
123, 4, 11syl2an 287 . 2 ((𝐴𝐶𝐵𝐷) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}))
132, 12mpd 13 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  {cab 2103  Vcvv 2660  wf 5089  (class class class)co 5742  𝑚 cmap 6510
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-map 6512
This theorem is referenced by:  mapval  6522  elmapg  6523  ixpconstg  6569  cnovex  12292  ispsmet  12419  cncfval  12655
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