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	⊢ (𝐹‘𝐴) ⊆ 𝐵

Proof of Theorem fpwfvss

Step	Hyp	Ref	Expression
1		fpwfvss.f	. . . 4 ⊢ 𝐹:𝐶⟶𝒫 𝐵
2	1	ffvelcdmi 7103	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵)
3	2	elpwid 4609	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
4	1	fdmi 6747	. . . . 5 ⊢ dom 𝐹 = 𝐶
5	4	eleq2i 2833	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶)
6		ndmfv 6941	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
7	5, 6	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅)
8		0ss 4400	. . 3 ⊢ ∅ ⊆ 𝐵
9	7, 8	eqsstrdi 4028	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
10	3, 9	pm2.61i 182	1 ⊢ (𝐹‘𝐴) ⊆ 𝐵

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)