Theorem mapsspw 8871

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 8869	. 2 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵)
2		pmsspw 8870	. 2 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
3	1, 2	sstri 3986	1 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Syntax hints:   ⊆ wss 3943  𝒫 cpw 4597   × cxp 5667  (class class class)co 7404   ↑_m cmap 8819   ↑_pm cpm 8820

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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

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-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  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-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-pm 8822

This theorem is referenced by:  mapfi  9347  rankmapu  9872  grumap  10802  wunfunc  17857  wunfuncOLD  17858