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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 4609 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵) |
| 4 | 1 | fdmi 6747 | . . . . 5 ⊢ dom 𝐹 = 𝐶 |
| 5 | 4 | eleq2i 2833 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶) |
| 6 | ndmfv 6941 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
| 7 | 5, 6 | sylnbir 331 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅) |
| 8 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
| 9 | 7, 8 | eqsstrdi 4028 | . 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 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 |
| This theorem is referenced by: (None) |
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