Theorem fpwfvss 43857

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 7029	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵)
3	2	elpwid 4551	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
4	1	fdmi 6673	. . . . 5 ⊢ dom 𝐹 = 𝐶
5	4	eleq2i 2829	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶)
6		ndmfv 6866	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
7	5, 6	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅)
8		0ss 4341	. . 3 ⊢ ∅ ⊆ 𝐵
9	7, 8	eqsstrdi 3967	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
10	3, 9	pm2.61i 182	1 ⊢ (𝐹‘𝐴) ⊆ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  dom cdm 5624  ⟶wf 6488  ‘cfv 6492

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by: (None)