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Theorem fpwfvss 42721
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 7078 . . 3 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) ∈ 𝒫 𝐡)
32elpwid 4606 . 2 (𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) βŠ† 𝐡)
41fdmi 6722 . . . . 5 dom 𝐹 = 𝐢
54eleq2i 2819 . . . 4 (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐢)
6 ndmfv 6919 . . . 4 (Β¬ 𝐴 ∈ dom 𝐹 β†’ (πΉβ€˜π΄) = βˆ…)
75, 6sylnbir 331 . . 3 (Β¬ 𝐴 ∈ 𝐢 β†’ (πΉβ€˜π΄) = βˆ…)
8 0ss 4391 . . 3 βˆ… βŠ† 𝐡
97, 8eqsstrdi 4031 . 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 3943  βˆ…c0 4317  π’« cpw 4597  dom cdm 5669  βŸΆwf 6532  β€˜cfv 6536
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by: (None)
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