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Mirrors > Home > MPE Home > Th. List > Mathboxes > fpwfvss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fpwfvss.f | ⊢ 𝐹:𝐶⟶𝒫 𝐵 |
Ref | Expression |
---|---|
fpwfvss | ⊢ (𝐹‘𝐴) ⊆ 𝐵 |
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 4614 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵) |
4 | 1 | fdmi 6748 | . . . . 5 ⊢ dom 𝐹 = 𝐶 |
5 | 4 | eleq2i 2831 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶) |
6 | ndmfv 6942 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | sylnbir 331 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅) |
8 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
9 | 7, 8 | eqsstrdi 4050 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵) |
10 | 3, 9 | pm2.61i 182 | 1 ⊢ (𝐹‘𝐴) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 |
This theorem is referenced by: (None) |
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