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Theorem fpwfvss 43425
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 4609 . 2 (𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
41fdmi 6747 . . . . 5 dom 𝐹 = 𝐶
54eleq2i 2833 . . . 4 (𝐴 ∈ dom 𝐹𝐴𝐶)
6 ndmfv 6941 . . . 4 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
75, 6sylnbir 331 . . 3 𝐴𝐶 → (𝐹𝐴) = ∅)
8 0ss 4400 . . 3 ∅ ⊆ 𝐵
97, 8eqsstrdi 4028 . 2 𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
103, 9pm2.61i 182 1 (𝐹𝐴) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  wss 3951  c0 4333  𝒫 cpw 4600  dom cdm 5685  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by: (None)
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