Theorem mapvalg 8447

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 8443	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
2	1	ancoms 462	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
3		elex 3416	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
4		elex 3416	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
5		feq3 6487	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑦⟶𝑥 ↔ 𝑓:𝑦⟶𝐴))
6	5	abbidv 2802	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ 𝑓:𝑦⟶𝑥} = {𝑓 ∣ 𝑓:𝑦⟶𝐴})
7		feq2 6486	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑓:𝑦⟶𝐴 ↔ 𝑓:𝐵⟶𝐴))
8	7	abbidv 2802	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∣ 𝑓:𝑦⟶𝐴} = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
9		df-map 8439	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
10	6, 8, 9	ovmpog 7324	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
11	10	3expia 1122	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}))
12	3, 4, 11	syl2an 599	. 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 399   = wceq 1542   ∈ wcel 2114  {cab 2716  Vcvv 3398  ⟶wf 6335  (class class class)co 7170   ↑_m cmap 8437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439

This theorem is referenced by:  mapval  8449  elmapg  8450  ixpconstg  8516  hashf1lem2  13908  efmndbasabf  18153  symgbasfi  18625  birthdaylem1  25689  birthdaylem2  25690  cnfex  42109