Theorem pmvalg 8497

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 3979	. . 3 ⊢ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴)
2		xpexg 7513	. . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐵 × 𝐴) ∈ V)
4	3	pwexd 5257	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐵 × 𝐴) ∈ V)
5		ssexg 5201	. . 3 ⊢ (({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ∈ V) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
6	1, 4, 5	sylancr 590	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
7		elex 3416	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
8		elex 3416	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
9		xpeq2 5557	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
10	9	pweqd 4518	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴))
11	10	rabeqdv 3385	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
12		xpeq1 5550	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
13	12	pweqd 4518	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴))
14	13	rabeqdv 3385	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
15		df-pm 8489	. . . . 5 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
16	11, 14, 15	ovmpog 7346	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
17	16	3expia 1123	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
18	7, 8, 17	syl2an 599	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
19	6, 18	mpd 15	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  {crab 3055  Vcvv 3398   ⊆ wss 3853  𝒫 cpw 4499   × cxp 5534  Fun wfun 6352  (class class class)co 7191   ↑_pm cpm 8487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-pm 8489

This theorem is referenced by:  elpmg  8502