Theorem pmsspw 8889

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 4286	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅)
2		fnpm 8838	. . . . . . . . 9 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6635	. . . . . . . 8 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7597	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅)
5	1, 4	nsyl2 142	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpmg 8847	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
7	5, 6	syl 18	. . . . 5 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
8	7	ibi 270	. . . 4 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)))
9	8	simprd 501	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
10		velpw 4562	. . 3 ⊢ (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
11	9, 10	sylibr 237	. 2 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
12	11	ssriv 3935	1 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   × cxp 5649  Fun wfun 6525  (class class class)co 7412   ↑_pm cpm 8832

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-pm 8834

This theorem is used by:  mapsspw  8890  wunpm  10791