Theorem fsetsspwxp 8830

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 6715	. . 3 ⊢ (𝑔:𝐴⟶𝐵 → 𝑔 ⊆ (𝐴 × 𝐵))
2		vex 3457	. . . 4 ⊢ 𝑔 ∈ V
3		feq1 6665	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵))
4	2, 3	elab 3638	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ↔ 𝑔:𝐴⟶𝐵)
5		velpw 4559	. . 3 ⊢ (𝑔 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑔 ⊆ (𝐴 × 𝐵))
6	1, 4, 5	3imtr4i 294	. 2 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓:𝐴⟶𝐵} → 𝑔 ∈ 𝒫 (𝐴 × 𝐵))
7	6	ssriv 3940	1 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)

Syntax hints:   ∈ wcel 2141  {cab 2739   ⊆ wss 3904  𝒫 cpw 4554   × cxp 5643  ⟶wf 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-fun 6519  df-fn 6520  df-f 6521

This theorem is referenced by:  sticksstones22  42749