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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 6524 | . . 3 ⊢ (𝑔:𝐴⟶𝐵 → 𝑔 ⊆ (𝐴 × 𝐵)) | |
2 | vex 3413 | . . . 4 ⊢ 𝑔 ∈ V | |
3 | feq1 6484 | . . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵)) | |
4 | 2, 3 | elab 3590 | . . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑔:𝐴⟶𝐵) |
5 | velpw 4502 | . . 3 ⊢ (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵)) | |
6 | 1, 4, 5 | 3imtr4i 295 | . 2 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵)) |
7 | 6 | ssriv 3898 | 1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 {cab 2735 ⊆ wss 3860 𝒫 cpw 4497 × cxp 5526 ⟶wf 6336 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-v 3411 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5037 df-opab 5099 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-fun 6342 df-fn 6343 df-f 6344 |
This theorem is referenced by: (None) |
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