Theorem mapvalg 8827

Description: The value of set exponentiation. (𝐴 ↑_m 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	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})

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.

Step	Hyp	Ref	Expression
1		mapex 8823	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
2	1	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
3		elex 3485	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
4		elex 3485	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
5		feq3 6691	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑦⟶𝑥 ↔ 𝑓:𝑦⟶𝐴))
6	5	abbidv 2793	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ 𝑓:𝑦⟶𝑥} = {𝑓 ∣ 𝑓:𝑦⟶𝐴})
7		feq2 6690	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑓:𝑦⟶𝐴 ↔ 𝑓:𝐵⟶𝐴))
8	7	abbidv 2793	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∣ 𝑓:𝑦⟶𝐴} = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
9		df-map 8819	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
10	6, 8, 9	ovmpog 7560	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
11	10	3expia 1118	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}))
12	3, 4, 11	syl2an 595	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}))
13	2, 12	mpd 15	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  Vcvv 3466  ⟶wf 6530  (class class class)co 7402   ↑_m cmap 8817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819

This theorem is referenced by:  mapval  8829  elmapg  8830  ixpconstg  8897  hashf1lem2  14415  efmndbasabf  18789  symgbasfi  19290  birthdaylem1  26802  birthdaylem2  26803  cnfex  44226