Theorem pmvalg 8020

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 3836	. . 3 ⊢ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴)
2		xpexg 7107	. . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 455	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐵 × 𝐴) ∈ V)
4		pwexg 4980	. . . 4 ⊢ ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ∈ V)
5	3, 4	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐵 × 𝐴) ∈ V)
6		ssexg 4938	. . 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 575	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
8		elex 3364	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
9		elex 3364	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
10		xpeq2 5269	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
11	10	pweqd 4302	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴))
12		rabeq 3342	. . . . . 6 ⊢ (𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴) → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
13	11, 12	syl 17	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
14		xpeq1 5263	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
15	14	pweqd 4302	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴))
16		rabeq 3342	. . . . . 6 ⊢ (𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴) → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
17	15, 16	syl 17	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
18		df-pm 8012	. . . . 5 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
19	13, 17, 18	ovmpt2g 6942	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
20	19	3expia 1114	. . 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 583	. 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 15	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065  Vcvv 3351   ⊆ wss 3723  𝒫 cpw 4297   × cxp 5247  Fun wfun 6025  (class class class)co 6793   ↑_pm cpm 8010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-pm 8012

This theorem is referenced by:  elpmg  8025