Theorem pmvalg 6933

Description: The value of the partial mapping operation. (𝐴 ↑_pm 𝐵) is the set of all partial functions that map from 𝐵 to 𝐴. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
pmvalg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem pmvalg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3333	. . 3 ⊢ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴)
2		xpexg 4889	. . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 268	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐵 × 𝐴) ∈ V)
4		pwexg 4317	. . . 4 ⊢ ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ∈ V)
5	3, 4	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐵 × 𝐴) ∈ V)
6		ssexg 4272	. . 3 ⊢ (({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ∈ V) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
7	1, 5, 6	sylancr 418	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
8		elex 2833	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
9		elex 2833	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
10		xpeq2 4789	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
11	10	pweqd 3693	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴))
12		rabeq 2813	. . . . . 6 ⊢ (𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴) → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
13	11, 12	syl 14	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
14		xpeq1 4788	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
15	14	pweqd 3693	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴))
16		rabeq 2813	. . . . . 6 ⊢ (𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴) → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
17	15, 16	syl 14	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
18		df-pm 6925	. . . . 5 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
19	13, 17, 18	ovmpog 6223	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
20	19	3expia 1236	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
21	8, 9, 20	syl2an 289	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
22	7, 21	mpd 13	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   × cxp 4772  Fun wfun 5371  (class class class)co 6085   ↑_pm cpm 6923

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pm 6925

This theorem is used by:  elpmg  6938