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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 7034 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ∈ 𝒫 𝐵) |
3 | 2 | elpwid 4569 | . 2 ⊢ (𝐴 ∈ 𝐶 → (𝐹‘𝐴) ⊆ 𝐵) |
4 | 1 | fdmi 6680 | . . . . 5 ⊢ dom 𝐹 = 𝐶 |
5 | 4 | eleq2i 2829 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝐶) |
6 | ndmfv 6877 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | sylnbir 330 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐶 → (𝐹‘𝐴) = ∅) |
8 | 0ss 4356 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
9 | 7, 8 | eqsstrdi 3998 | . 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 1541 ∈ wcel 2106 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 dom cdm 5633 ⟶wf 6492 ‘cfv 6496 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 |
This theorem is referenced by: (None) |
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