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Theorem xpnum 8323
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 8317 . 2  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
2 isnum2 8317 . 2  |-  ( B  e.  dom  card  <->  E. y  e.  On  y  ~~  B
)
3 reeanv 3022 . . 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 7178 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  .o  y
)  e.  On )
54adantr 463 . . . . . 6  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  (
x  .o  y )  e.  On )
6 omxpen 7612 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  .o  y
)  ~~  ( x  X.  y ) )
7 xpen 7673 . . . . . . 7  |-  ( ( x  ~~  A  /\  y  ~~  B )  -> 
( x  X.  y
)  ~~  ( A  X.  B ) )
8 entr 7560 . . . . . . 7  |-  ( ( ( x  .o  y
)  ~~  ( x  X.  y )  /\  (
x  X.  y ) 
~~  ( A  X.  B ) )  -> 
( x  .o  y
)  ~~  ( A  X.  B ) )
96, 7, 8syl2an 475 . . . . . 6  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  (
x  .o  y ) 
~~  ( A  X.  B ) )
10 isnumi 8318 . . . . . 6  |-  ( ( ( x  .o  y
)  e.  On  /\  ( x  .o  y
)  ~~  ( A  X.  B ) )  -> 
( A  X.  B
)  e.  dom  card )
115, 9, 10syl2anc 659 . . . . 5  |-  ( ( ( x  e.  On  /\  y  e.  On )  /\  ( x  ~~  A  /\  y  ~~  B
) )  ->  ( A  X.  B )  e. 
dom  card )
1211ex 432 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  ~~  A  /\  y  ~~  B
)  ->  ( A  X.  B )  e.  dom  card ) )
1312rexlimivv 2951 . . 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 477 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 367    e. wcel 1823   E.wrex 2805   class class class wbr 4439   Oncon0 4867    X. cxp 4986   dom cdm 4988  (class class class)co 6270    .o comu 7120    ~~ cen 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-en 7510  df-dom 7511  df-card 8311
This theorem is referenced by:  iunfictbso  8486  znnen  14030  qnnen  14031  ptcmplem2  20719  finixpnum  30278  isnumbasgrplem2  31294
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