Theorem mapvalg 8861

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 7954	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
2	1	ancoms 464	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V)
3		elex 3471	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
4		elex 3471	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
5		feq3 6692	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑦⟶𝑥 ↔ 𝑓:𝑦⟶𝐴))
6	5	abbidv 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ 𝑓:𝑦⟶𝑥} = {𝑓 ∣ 𝑓:𝑦⟶𝐴})
7		feq2 6691	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑓:𝑦⟶𝐴 ↔ 𝑓:𝐵⟶𝐴))
8	7	abbidv 2826	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∣ 𝑓:𝑦⟶𝐴} = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
9		df-map 8854	. . . . 5 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
10	6, 8, 9	ovmpog 7581	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
11	10	3expia 1139	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}))
12	3, 4, 11	syl2an 608	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝑓 ∣ 𝑓:𝐵⟶𝐴} ∈ V → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}))
13	2, 12	mpd 16	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450  ⟶wf 6537  (class class class)co 7422   ↑_m cmap 8852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-map 8854

This theorem is used by:  mapval  8863  elmapg  8864  ixpconstg  8939  hashf1lem2  14606  efmndbasabf  19075  symgbasfi  19600  birthdaylem1  27286  birthdaylem2  27287  cnfex  46152