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Theorem cardiun 8380
Description: The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)
Assertion
Ref Expression
cardiun  |-  ( A  e.  V  ->  ( A. x  e.  A  ( card `  B )  =  B  ->  ( card `  U_ x  e.  A  B )  =  U_ x  e.  A  B
) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem cardiun
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abrexexg 6774 . . . . . 6  |-  ( A  e.  V  ->  { z  |  E. x  e.  A  z  =  (
card `  B ) }  e.  _V )
2 vex 3112 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2461 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  ( card `  B )  <->  y  =  ( card `  B )
) )
43rexbidv 2968 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. x  e.  A  z  =  ( card `  B )  <->  E. x  e.  A  y  =  ( card `  B )
) )
52, 4elab 3246 . . . . . . . 8  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  <->  E. x  e.  A  y  =  ( card `  B )
)
6 cardidm 8357 . . . . . . . . . 10  |-  ( card `  ( card `  B
) )  =  (
card `  B )
7 fveq2 5872 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  (
card `  ( card `  B ) ) )
8 id 22 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  y  =  ( card `  B )
)
96, 7, 83eqtr4a 2524 . . . . . . . . 9  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  y )
109rexlimivw 2946 . . . . . . . 8  |-  ( E. x  e.  A  y  =  ( card `  B
)  ->  ( card `  y )  =  y )
115, 10sylbi 195 . . . . . . 7  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  ->  (
card `  y )  =  y )
1211rgen 2817 . . . . . 6  |-  A. y  e.  { z  |  E. x  e.  A  z  =  ( card `  B
) }  ( card `  y )  =  y
13 carduni 8379 . . . . . 6  |-  ( { z  |  E. x  e.  A  z  =  ( card `  B ) }  e.  _V  ->  ( A. y  e.  {
z  |  E. x  e.  A  z  =  ( card `  B ) }  ( card `  y
)  =  y  -> 
( card `  U. { z  |  E. x  e.  A  z  =  (
card `  B ) } )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) } ) )
141, 12, 13mpisyl 18 . . . . 5  |-  ( A  e.  V  ->  ( card `  U. { z  |  E. x  e.  A  z  =  (
card `  B ) } )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) } )
15 fvex 5882 . . . . . . 7  |-  ( card `  B )  e.  _V
1615dfiun2 4366 . . . . . 6  |-  U_ x  e.  A  ( card `  B )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) }
1716fveq2i 5875 . . . . 5  |-  ( card `  U_ x  e.  A  ( card `  B )
)  =  ( card `  U. { z  |  E. x  e.  A  z  =  ( card `  B ) } )
1814, 17, 163eqtr4g 2523 . . . 4  |-  ( A  e.  V  ->  ( card `  U_ x  e.  A  ( card `  B
) )  =  U_ x  e.  A  ( card `  B ) )
1918adantr 465 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  ( card `  U_ x  e.  A  ( card `  B
) )  =  U_ x  e.  A  ( card `  B ) )
20 iuneq2 4349 . . . . 5  |-  ( A. x  e.  A  ( card `  B )  =  B  ->  U_ x  e.  A  ( card `  B
)  =  U_ x  e.  A  B )
2120adantl 466 . . . 4  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  U_ x  e.  A  ( card `  B )  =  U_ x  e.  A  B
)
2221fveq2d 5876 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  ( card `  U_ x  e.  A  ( card `  B
) )  =  (
card `  U_ x  e.  A  B ) )
2319, 22, 213eqtr3d 2506 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  ( card `  U_ x  e.  A  B )  = 
U_ x  e.  A  B )
2423ex 434 1  |-  ( A  e.  V  ->  ( A. x  e.  A  ( card `  B )  =  B  ->  ( card `  U_ x  e.  A  B )  =  U_ x  e.  A  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   _Vcvv 3109   U.cuni 4251   U_ciun 4332   ` cfv 5594   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-card 8337
This theorem is referenced by:  alephcard  8468
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