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Theorem fsetsspwxp 8853
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 6745 . . 3 (𝑔:𝐴𝐵𝑔 ⊆ (𝐴 × 𝐵))
2 vex 3477 . . . 4 𝑔 ∈ V
3 feq1 6698 . . . 4 (𝑓 = 𝑔 → (𝑓:𝐴𝐵𝑔:𝐴𝐵))
42, 3elab 3668 . . 3 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑔:𝐴𝐵)
5 velpw 4607 . . 3 (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵))
61, 4, 53imtr4i 292 . 2 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵))
76ssriv 3986 1 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  {cab 2708  wss 3948  𝒫 cpw 4602   × cxp 5674  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  sticksstones22  41451
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