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Theorem fpwfvss 43402
Description: Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.)
Hypothesis
Ref Expression
fpwfvss.f 𝐹:𝐶⟶𝒫 𝐵
Assertion
Ref Expression
fpwfvss (𝐹𝐴) ⊆ 𝐵

Proof of Theorem fpwfvss
StepHypRef Expression
1 fpwfvss.f . . . 4 𝐹:𝐶⟶𝒫 𝐵
21ffvelcdmi 7103 . . 3 (𝐴𝐶 → (𝐹𝐴) ∈ 𝒫 𝐵)
32elpwid 4614 . 2 (𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
41fdmi 6748 . . . . 5 dom 𝐹 = 𝐶
54eleq2i 2831 . . . 4 (𝐴 ∈ dom 𝐹𝐴𝐶)
6 ndmfv 6942 . . . 4 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
75, 6sylnbir 331 . . 3 𝐴𝐶 → (𝐹𝐴) = ∅)
8 0ss 4406 . . 3 ∅ ⊆ 𝐵
97, 8eqsstrdi 4050 . 2 𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
103, 9pm2.61i 182 1 (𝐹𝐴) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  wss 3963  c0 4339  𝒫 cpw 4605  dom cdm 5689  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by: (None)
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