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Mirrors > Home > MPE Home > Th. List > fsetsspwxp | Structured version Visualization version GIF version |
Description: The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.) |
Ref | Expression |
---|---|
fsetsspwxp | ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssxp 6746 | . . 3 ⊢ (𝑔:𝐴⟶𝐵 → 𝑔 ⊆ (𝐴 × 𝐵)) | |
2 | vex 3467 | . . . 4 ⊢ 𝑔 ∈ V | |
3 | feq1 6698 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵)) | |
4 | 2, 3 | elab 3659 | . . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑔:𝐴⟶𝐵) |
5 | velpw 4603 | . . 3 ⊢ (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵)) | |
6 | 1, 4, 5 | 3imtr4i 291 | . 2 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵)) |
7 | 6 | ssriv 3976 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2702 ⊆ wss 3939 𝒫 cpw 4598 × cxp 5670 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: sticksstones22 41696 |
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