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Theorem pmsspw 6685
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 6653 . . . . . . 7 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
21elmpocl 6071 . . . . . 6 (𝑓 ∈ (𝐴pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 elpmg 6666 . . . . . 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 176 . . . 4 (𝑓 ∈ (𝐴pm 𝐵) → (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)))
65simprd 114 . . 3 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
7 velpw 3584 . . 3 (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
86, 7sylibr 134 . 2 (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
98ssriv 3161 1 (𝐴pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2148  {crab 2459  Vcvv 2739  wss 3131  𝒫 cpw 3577   × cxp 4626  Fun wfun 5212  (class class class)co 5877  pm cpm 6651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pm 6653
This theorem is referenced by:  mapsspw  6686
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