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Theorem mapvalg 6658
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 6654 . . 3 ((𝐵𝐷𝐴𝐶) → {𝑓𝑓:𝐵𝐴} ∈ V)
21ancoms 268 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐵𝐴} ∈ V)
3 elex 2749 . . 3 (𝐴𝐶𝐴 ∈ V)
4 elex 2749 . . 3 (𝐵𝐷𝐵 ∈ V)
5 feq3 5351 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑦𝑥𝑓:𝑦𝐴))
65abbidv 2295 . . . . 5 (𝑥 = 𝐴 → {𝑓𝑓:𝑦𝑥} = {𝑓𝑓:𝑦𝐴})
7 feq2 5350 . . . . . 6 (𝑦 = 𝐵 → (𝑓:𝑦𝐴𝑓:𝐵𝐴))
87abbidv 2295 . . . . 5 (𝑦 = 𝐵 → {𝑓𝑓:𝑦𝐴} = {𝑓𝑓:𝐵𝐴})
9 df-map 6650 . . . . 5 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
106, 8, 9ovmpog 6009 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓𝑓:𝐵𝐴} ∈ V) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
11103expia 1205 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}))
123, 4, 11syl2an 289 . 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 104   = wceq 1353  wcel 2148  {cab 2163  Vcvv 2738  wf 5213  (class class class)co 5875  𝑚 cmap 6648
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-map 6650
This theorem is referenced by:  mapval  6660  elmapg  6661  ixpconstg  6707  ptex  12713  cnovex  13699  ispsmet  13826  cncfval  14062
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