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Theorem pmsspw 6545
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 6513 . . . . . . 7 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
21elmpocl 5936 . . . . . 6 (𝑓 ∈ (𝐴pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 elpmg 6526 . . . . . 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 3487 . . 3 (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
86, 7sylibr 133 . 2 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
98ssriv 3071 1 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1465  {crab 2397  Vcvv 2660  wss 3041  𝒫 cpw 3480   × cxp 4507  Fun wfun 5087  (class class class)co 5742  pm cpm 6511
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pm 6513
This theorem is referenced by:  mapsspw  6546
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