Theorem fsetsspwxp 8838

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 6723	. . 3 ⊢ (𝑔:𝐴⟶𝐵 → 𝑔 ⊆ (𝐴 × 𝐵))
2		vex 3461	. . . 4 ⊢ 𝑔 ∈ V
3		feq1 6673	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
4	2, 3	elab 3641	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑔:𝐴⟶𝐵)
5		velpw 4563	. . 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 3943	1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)

Syntax hints:   ∈ wcel 2145  {cab 2743   ⊆ wss 3907  𝒫 cpw 4558   × cxp 5649  ⟶wf 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6527  df-fn 6528  df-f 6529

This theorem is referenced by:  sticksstones22  42792