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Theorem fpwfvss 41758
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 7039 . . 3 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) ∈ 𝒫 𝐡)
32elpwid 4574 . 2 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) βŠ† 𝐡)
41fdmi 6685 . . . . 5 dom 𝐹 = 𝐢
54eleq2i 2830 . . . 4 (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐢)
6 ndmfv 6882 . . . 4 (Β¬ 𝐴 ∈ dom 𝐹 β†’ (πΉβ€˜π΄) = βˆ…)
75, 6sylnbir 331 . . 3 (Β¬ 𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) = βˆ…)
8 0ss 4361 . . 3 βˆ… βŠ† 𝐡
97, 8eqsstrdi 4003 . 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 1542   ∈ wcel 2107   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  dom cdm 5638  βŸΆwf 6497  β€˜cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509
This theorem is referenced by: (None)
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