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Theorem fsetsspwxp 8849
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 6739 . . 3 (𝑔:𝐴𝐵𝑔 ⊆ (𝐴 × 𝐵))
2 vex 3472 . . . 4 𝑔 ∈ V
3 feq1 6692 . . . 4 (𝑓 = 𝑔 → (𝑓:𝐴𝐵𝑔:𝐴𝐵))
42, 3elab 3663 . . 3 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} ↔ 𝑔:𝐴𝐵)
5 velpw 4602 . . 3 (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵))
61, 4, 53imtr4i 292 . 2 (𝑔 ∈ {𝑓𝑓:𝐴𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵))
76ssriv 3981 1 {𝑓𝑓:𝐴𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2703  wss 3943  𝒫 cpw 4597   × cxp 5667  wf 6533
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6539  df-fn 6540  df-f 6541
This theorem is referenced by:  sticksstones22  41546
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