Theorem fpwfvss 42153

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 7085	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵)
3	2	elpwid 4611	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
4	1	fdmi 6729	. . . . 5 ⊢ dom 𝐹 = 𝐶
5	4	eleq2i 2825	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶)
6		ndmfv 6926	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
7	5, 6	sylnbir 330	. . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅)
8		0ss 4396	. . 3 ⊢ ∅ ⊆ 𝐵
9	7, 8	eqsstrdi 4036	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
10	3, 9	pm2.61i 182	1 ⊢ (𝐹‘𝐴) ⊆ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  dom cdm 5676  ⟶wf 6539  ‘cfv 6543

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by: (None)