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Theorem fsetsspwxp 8794
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.)
Assertion
Ref Expression
fsetsspwxp {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fsetsspwxp
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fssxp 6697 . . 3 (𝑔:𝐴𝐵𝑔 ⊆ (𝐴 × 𝐵))
2 vex 3448 . . . 4 𝑔 ∈ V
3 feq1 6650 . . . 4 (𝑓 = 𝑔 → (𝑓:𝐴𝐵𝑔:𝐴𝐵))
42, 3elab 3631 . . 3 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑔:𝐴𝐵)
5 velpw 4566 . . 3 (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵))
61, 4, 53imtr4i 292 . 2 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵))
76ssriv 3949 1 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {cab 2710  wss 3911  𝒫 cpw 4561   × cxp 5632  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  sticksstones22  40622
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