Theorem fpwfvss 43863

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 7031	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵)
3	2	elpwid 4545	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
4	1	fdmi 6673	. . . . 5 ⊢ dom 𝐹 = 𝐶
5	4	eleq2i 2832	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶)
6		ndmfv 6866	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
7	5, 6	sylnbir 332	. . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅)
8		0ss 4335	. . 3 ⊢ ∅ ⊆ 𝐵
9	7, 8	eqsstrdi 3966	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
10	3, 9	pm2.61i 183	1 ⊢ (𝐹‘𝐴) ⊆ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  dom cdm 5625  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by: (None)