Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fpwfvss Structured version   Visualization version   GIF version

Theorem fpwfvss 41674
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 7034 . . 3 (𝐴𝐶 → (𝐹𝐴) ∈ 𝒫 𝐵)
32elpwid 4569 . 2 (𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
41fdmi 6680 . . . . 5 dom 𝐹 = 𝐶
54eleq2i 2829 . . . 4 (𝐴 ∈ dom 𝐹𝐴𝐶)
6 ndmfv 6877 . . . 4 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
75, 6sylnbir 330 . . 3 𝐴𝐶 → (𝐹𝐴) = ∅)
8 0ss 4356 . . 3 ∅ ⊆ 𝐵
97, 8eqsstrdi 3998 . 2 𝐴𝐶 → (𝐹𝐴) ⊆ 𝐵)
103, 9pm2.61i 182 1 (𝐹𝐴) ⊆ 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  wss 3910  c0 4282  𝒫 cpw 4560  dom cdm 5633  wf 6492  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator