Metamath Proof Explorer

Theorem enrelmap

Description: The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. See rfovf1od for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021)

Ref Expression
Assertion enrelmap 
|- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ( ~P B ^m A ) )

Proof

Step Hyp Ref Expression
1 xpcomeng 
 |-  ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )
2 pwen 
 |-  ( ( A X. B ) ~~ ( B X. A ) -> ~P ( A X. B ) ~~ ~P ( B X. A ) )
3 1 2 syl 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ~P ( B X. A ) )
4 xpexg 
 |-  ( ( B e. W /\ A e. V ) -> ( B X. A ) e. _V )
5 4 ancoms 
 |-  ( ( A e. V /\ B e. W ) -> ( B X. A ) e. _V )
6 pw2eng 
 |-  ( ( B X. A ) e. _V -> ~P ( B X. A ) ~~ ( 2o ^m ( B X. A ) ) )
7 5 6 syl 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( B X. A ) ~~ ( 2o ^m ( B X. A ) ) )
8 entr 
 |-  ( ( ~P ( A X. B ) ~~ ~P ( B X. A ) /\ ~P ( B X. A ) ~~ ( 2o ^m ( B X. A ) ) ) -> ~P ( A X. B ) ~~ ( 2o ^m ( B X. A ) ) )
9 3 7 8 syl2anc 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ( 2o ^m ( B X. A ) ) )
10 pw2eng 
 |-  ( B e. W -> ~P B ~~ ( 2o ^m B ) )
11 enrefg 
 |-  ( A e. V -> A ~~ A )
12 mapen 
 |-  ( ( ~P B ~~ ( 2o ^m B ) /\ A ~~ A ) -> ( ~P B ^m A ) ~~ ( ( 2o ^m B ) ^m A ) )
13 10 11 12 syl2anr 
 |-  ( ( A e. V /\ B e. W ) -> ( ~P B ^m A ) ~~ ( ( 2o ^m B ) ^m A ) )
14 2on 
 |-  2o e. On
15 simpr 
 |-  ( ( A e. V /\ B e. W ) -> B e. W )
16 simpl 
 |-  ( ( A e. V /\ B e. W ) -> A e. V )
17 mapxpen 
 |-  ( ( 2o e. On /\ B e. W /\ A e. V ) -> ( ( 2o ^m B ) ^m A ) ~~ ( 2o ^m ( B X. A ) ) )
18 14 15 16 17 mp3an2i 
 |-  ( ( A e. V /\ B e. W ) -> ( ( 2o ^m B ) ^m A ) ~~ ( 2o ^m ( B X. A ) ) )
19 entr 
 |-  ( ( ( ~P B ^m A ) ~~ ( ( 2o ^m B ) ^m A ) /\ ( ( 2o ^m B ) ^m A ) ~~ ( 2o ^m ( B X. A ) ) ) -> ( ~P B ^m A ) ~~ ( 2o ^m ( B X. A ) ) )
20 13 18 19 syl2anc 
 |-  ( ( A e. V /\ B e. W ) -> ( ~P B ^m A ) ~~ ( 2o ^m ( B X. A ) ) )
21 20 ensymd 
 |-  ( ( A e. V /\ B e. W ) -> ( 2o ^m ( B X. A ) ) ~~ ( ~P B ^m A ) )
22 entr 
 |-  ( ( ~P ( A X. B ) ~~ ( 2o ^m ( B X. A ) ) /\ ( 2o ^m ( B X. A ) ) ~~ ( ~P B ^m A ) ) -> ~P ( A X. B ) ~~ ( ~P B ^m A ) )
23 9 21 22 syl2anc 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) ~~ ( ~P B ^m A ) )