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Theorem fpwfvss 42873
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:πΆβŸΆπ’« 𝐡
Assertion
Ref Expression
fpwfvss (πΉβ€˜π΄) βŠ† 𝐡

Proof of Theorem fpwfvss
StepHypRef Expression
1 fpwfvss.f . . . 4 𝐹:πΆβŸΆπ’« 𝐡
21ffvelcdmi 7098 . . 3 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) ∈ 𝒫 𝐡)
32elpwid 4615 . 2 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) βŠ† 𝐡)
41fdmi 6739 . . . . 5 dom 𝐹 = 𝐢
54eleq2i 2821 . . . 4 (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐢)
6 ndmfv 6937 . . . 4 (Β¬ 𝐴 ∈ dom 𝐹 β†’ (πΉβ€˜π΄) = βˆ…)
75, 6sylnbir 330 . . 3 (Β¬ 𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) = βˆ…)
8 0ss 4400 . . 3 βˆ… βŠ† 𝐡
97, 8eqsstrdi 4036 . 2 (Β¬ 𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) βŠ† 𝐡)
103, 9pm2.61i 182 1 (πΉβ€˜π΄) βŠ† 𝐡
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  dom cdm 5682  βŸΆwf 6549  β€˜cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561
This theorem is referenced by: (None)
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