Theorem mapsspw 8916

Description: Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
mapsspw	⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Proof of Theorem mapsspw

Step	Hyp	Ref	Expression
1		mapsspm 8914	. 2 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵)
2		pmsspw 8915	. 2 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
3	1, 2	sstri 4004	1 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Syntax hints:   ⊆ wss 3962  𝒫 cpw 4604   × cxp 5686  (class class class)co 7430   ↑_m cmap 8864   ↑_pm cpm 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-pm 8867

This theorem is referenced by:  mapfi  9385  rankmapu  9915  grumap  10845  wunfunc  17951  wunfuncOLD  17952