Theorem pmvalg 8861

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 4005	. . 3 ⊢ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴)
2		xpexg 7765	. . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴 ∈ 𝐶) → (𝐵 × 𝐴) ∈ V)
3	2	ancoms 464	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐵 × 𝐴) ∈ V)
4	3	pwexd 5340	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐵 × 𝐴) ∈ V)
5		ssexg 5280	. . 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 599	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
7		elex 3471	. . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
8		elex 3471	. . 3 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
9		xpeq2 5671	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
10	9	pweqd 4573	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴))
11	10	rabeqdv 3427	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
12		xpeq1 5664	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
13	12	pweqd 4573	. . . . . 6 ⊢ (𝑦 = 𝐵 → 𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴))
14	13	rabeqdv 3427	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
15		df-pm 8854	. . . . 5 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
16	11, 14, 15	ovmpog 7580	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
17	16	3expia 1139	. . 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 608	. 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 16	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556   × cxp 5648  Fun wfun 6533  (class class class)co 7421   ↑_pm cpm 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 7752

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 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-pm 8854

This theorem is used by:  elpmg  8867