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Theorem pmsspw 6588
 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 variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pm 6556 . . . . . . 7 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
21elmpocl 5979 . . . . . 6 (𝑓 ∈ (𝐴pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 elpmg 6569 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
42, 3syl 14 . . . . 5 (𝑓 ∈ (𝐴pm 𝐵) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
54ibi 175 . . . 4 (𝑓 ∈ (𝐴pm 𝐵) → (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)))
65simprd 113 . . 3 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
7 velpw 3523 . . 3 (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
86, 7sylibr 133 . 2 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
98ssriv 3107 1 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∈ wcel 1481  {crab 2421  Vcvv 2690   ⊆ wss 3077  𝒫 cpw 3516   × cxp 4548  Fun wfun 5128  (class class class)co 5785   ↑pm cpm 6554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-iota 5099  df-fun 5136  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790  df-pm 6556 This theorem is referenced by:  mapsspw  6589
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