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Theorem xpnum 8119
Description: The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xpnum  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  X.  B )  e.  dom  card )

Proof of Theorem xpnum
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnum2 8113 . 2  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
2 isnum2 8113 . 2  |-  ( B  e.  dom  card  <->  E. y  e.  On  y  ~~  B
)
3 reeanv 2886 . . 3  |-  ( E. x  e.  On  E. y  e.  On  (
x  ~~  A  /\  y  ~~  B )  <->  ( E. x  e.  On  x  ~~  A  /\  E. y  e.  On  y  ~~  B
) )
4 omcl 6974 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  .o  y
)  e.  On )
54adantr 465 . . . . . 6  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  (
x  .o  y )  e.  On )
6 omxpen 7411 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  .o  y
)  ~~  ( x  X.  y ) )
7 xpen 7472 . . . . . . 7  |-  ( ( x  ~~  A  /\  y  ~~  B )  -> 
( x  X.  y
)  ~~  ( A  X.  B ) )
8 entr 7359 . . . . . . 7  |-  ( ( ( x  .o  y
)  ~~  ( x  X.  y )  /\  (
x  X.  y ) 
~~  ( A  X.  B ) )  -> 
( x  .o  y
)  ~~  ( A  X.  B ) )
96, 7, 8syl2an 477 . . . . . 6  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  (
x  .o  y ) 
~~  ( A  X.  B ) )
10 isnumi 8114 . . . . . 6  |-  ( ( ( x  .o  y
)  e.  On  /\  ( x  .o  y
)  ~~  ( A  X.  B ) )  -> 
( A  X.  B
)  e.  dom  card )
115, 9, 10syl2anc 661 . . . . 5  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  ( A  X.  B )  e. 
dom  card )
1211ex 434 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  ~~  A  /\  y  ~~  B
)  ->  ( A  X.  B )  e.  dom  card ) )
1312rexlimivv 2844 . . 3  |-  ( E. x  e.  On  E. y  e.  On  (
x  ~~  A  /\  y  ~~  B )  -> 
( A  X.  B
)  e.  dom  card )
143, 13sylbir 213 . 2  |-  ( ( E. x  e.  On  x  ~~  A  /\  E. y  e.  On  y  ~~  B )  ->  ( A  X.  B )  e. 
dom  card )
151, 2, 14syl2anb 479 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  X.  B )  e.  dom  card )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   E.wrex 2714   class class class wbr 4290   Oncon0 4717    X. cxp 4836   dom cdm 4838  (class class class)co 6089    .o comu 6916    ~~ cen 7305   cardccrd 8103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-omul 6923  df-er 7099  df-en 7309  df-dom 7310  df-card 8107
This theorem is referenced by:  iunfictbso  8282  znnen  13493  qnnen  13494  ptcmplem2  19623  finixpnum  28411  isnumbasgrplem2  29457
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