Theorem fpwfvss 43952

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 7060	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵)
3	2	elpwid 4563	. 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
4	1	fdmi 6699	. . . . 5 ⊢ dom 𝐹 = 𝐶
5	4	eleq2i 2853	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶)
6		ndmfv 6895	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
7	5, 6	sylnbir 333	. . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅)
8		0ss 4353	. . 3 ⊢ ∅ ⊆ 𝐵
9	7, 8	eqsstrdi 3980	. 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵)
10	3, 9	pm2.61i 183	1 ⊢ (𝐹‘𝐴) ⊆ 𝐵

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  dom cdm 5645  ⟶wf 6513  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525

This theorem is referenced by: (None)