Theorem fsetsspwxp 8894

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	⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem fsetsspwxp

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fssxp 6762	. . 3 ⊢ (𝑔:𝐴⟶𝐵 → 𝑔 ⊆ (𝐴 × 𝐵))
2		vex 3483	. . . 4 ⊢ 𝑔 ∈ V
3		feq1 6715	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
4	2, 3	elab 3678	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑔:𝐴⟶𝐵)
5		velpw 4604	. . 3 ⊢ (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵))
6	1, 4, 5	3imtr4i 292	. 2 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵))
7	6	ssriv 3986	1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)

Syntax hints:   ∈ wcel 2107  {cab 2713   ⊆ wss 3950  𝒫 cpw 4599   × cxp 5682  ⟶wf 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-f 6564

This theorem is referenced by:  sticksstones22  42170