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Theorem pmsspw 8416
 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 4272 . . . . . . 7 (𝑓 ∈ (𝐴pm 𝐵) → ¬ (𝐴pm 𝐵) = ∅)
2 fnpm 8389 . . . . . . . . 9 pm Fn (V × V)
3 fndm 6428 . . . . . . . . 9 ( ↑pm Fn (V × V) → dom ↑pm = (V × V))
42, 3ax-mp 5 . . . . . . . 8 dom ↑pm = (V × V)
54ndmov 7307 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴pm 𝐵) = ∅)
61, 5nsyl2 143 . . . . . 6 (𝑓 ∈ (𝐴pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 elpmg 8397 . . . . . 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 270 . . . 4 (𝑓 ∈ (𝐴pm 𝐵) → (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)))
109simprd 499 . . 3 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
11 velpw 4517 . . 3 (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
1210, 11sylibr 237 . 2 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
1312ssriv 3947 1 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3471   ⊆ wss 3910  ∅c0 4266  𝒫 cpw 4512   × cxp 5526  dom cdm 5528  Fun wfun 6322   Fn wfn 6323  (class class class)co 7130   ↑pm cpm 8382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-pm 8384 This theorem is referenced by:  mapsspw  8417  wunpm  10124
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