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Theorem pmsspw 7778
 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 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Proof of Theorem pmsspw
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0i 3879 . . . . . . 7 (𝑓 ∈ (𝐴pm 𝐵) → ¬ (𝐴pm 𝐵) = ∅)
2 fnpm 7752 . . . . . . . . 9 pm Fn (V × V)
3 fndm 5904 . . . . . . . . 9 ( ↑pm Fn (V × V) → dom ↑pm = (V × V))
42, 3ax-mp 5 . . . . . . . 8 dom ↑pm = (V × V)
54ndmov 6716 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴pm 𝐵) = ∅)
61, 5nsyl2 141 . . . . . 6 (𝑓 ∈ (𝐴pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 elpmg 7759 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
86, 7syl 17 . . . . 5 (𝑓 ∈ (𝐴pm 𝐵) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
98ibi 255 . . . 4 (𝑓 ∈ (𝐴pm 𝐵) → (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)))
109simprd 478 . . 3 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
11 selpw 4115 . . 3 (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
1210, 11sylibr 223 . 2 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
1312ssriv 3572 1 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   × cxp 5036  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  (class class class)co 6549   ↑pm cpm 7745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-pm 7747 This theorem is referenced by:  mapsspw  7779  wunpm  9426
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