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Theorem mapvalg 7726
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 7722 . . 3 ((𝐵𝐷𝐴𝐶) → {𝑓𝑓:𝐵𝐴} ∈ V)
21ancoms 467 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓𝑓:𝐵𝐴} ∈ V)
3 elex 3179 . . 3 (𝐴𝐶𝐴 ∈ V)
4 elex 3179 . . 3 (𝐵𝐷𝐵 ∈ V)
5 feq3 5922 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑦𝑥𝑓:𝑦𝐴))
65abbidv 2722 . . . . 5 (𝑥 = 𝐴 → {𝑓𝑓:𝑦𝑥} = {𝑓𝑓:𝑦𝐴})
7 feq2 5921 . . . . . 6 (𝑦 = 𝐵 → (𝑓:𝑦𝐴𝑓:𝐵𝐴))
87abbidv 2722 . . . . 5 (𝑦 = 𝐵 → {𝑓𝑓:𝑦𝐴} = {𝑓𝑓:𝐵𝐴})
9 df-map 7718 . . . . 5 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
106, 8, 9ovmpt2g 6666 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓𝑓:𝐵𝐴} ∈ V) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
11103expia 1258 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓𝑓:𝐵𝐴} ∈ V → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}))
123, 4, 11syl2an 492 . 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 382   = wceq 1474  wcel 1975  {cab 2590  Vcvv 3167  wf 5781  (class class class)co 6522  𝑚 cmap 7716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-fv 5793  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-map 7718
This theorem is referenced by:  mapval  7728  elmapg  7729  ixpconstg  7775  hashf1lem2  13044  symgbasfi  17570  birthdaylem1  24390  birthdaylem2  24391  cnfex  38008
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