Theorem pmsspw 8801

Description: Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
pmsspw	⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Proof of Theorem pmsspw

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4290	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅)
2		fnpm 8758	. . . . . . . . 9 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6585	. . . . . . . 8 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7530	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpmg 8767	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
8	7	ibi 267	. . . 4 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)))
9	8	simprd 495	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
10		velpw 4555	. . 3 ⊢ (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
11	9, 10	sylibr 234	. 2 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
12	11	ssriv 3938	1 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550   × cxp 5614  Fun wfun 6475  (class class class)co 7346   ↑_pm cpm 8751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-pm 8753

This theorem is referenced by:  mapsspw  8802  wunpm  10613