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Theorem cardiun 7615
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 5764 . . . . . 6  |-  ( A  e.  V  ->  { z  |  E. x  e.  A  z  =  (
card `  B ) }  e.  _V )
2 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2289 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  ( card `  B )  <->  y  =  ( card `  B )
) )
43rexbidv 2564 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. x  e.  A  z  =  ( card `  B )  <->  E. x  e.  A  y  =  ( card `  B )
) )
52, 4elab 2914 . . . . . . . 8  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  <->  E. x  e.  A  y  =  ( card `  B )
)
6 cardidm 7592 . . . . . . . . . 10  |-  ( card `  ( card `  B
) )  =  (
card `  B )
7 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  (
card `  ( card `  B ) ) )
8 id 19 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  y  =  ( card `  B )
)
96, 7, 83eqtr4a 2341 . . . . . . . . 9  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  y )
109rexlimivw 2663 . . . . . . . 8  |-  ( E. x  e.  A  y  =  ( card `  B
)  ->  ( card `  y )  =  y )
115, 10sylbi 187 . . . . . . 7  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  ->  (
card `  y )  =  y )
1211rgen 2608 . . . . . 6  |-  A. y  e.  { z  |  E. x  e.  A  z  =  ( card `  B
) }  ( card `  y )  =  y
13 carduni 7614 . . . . . 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, 13ee10 1366 . . . . 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 5539 . . . . . . 7  |-  ( card `  B )  e.  _V
1615dfiun2 3937 . . . . . 6  |-  U_ x  e.  A  ( card `  B )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) }
1716fveq2i 5528 . . . . 5  |-  ( card `  U_ x  e.  A  ( card `  B )
)  =  ( card `  U. { z  |  E. x  e.  A  z  =  ( card `  B ) } )
1814, 17, 163eqtr4g 2340 . . . 4  |-  ( A  e.  V  ->  ( card `  U_ x  e.  A  ( card `  B
) )  =  U_ x  e.  A  ( card `  B ) )
1918adantr 451 . . 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 3921 . . . . 5  |-  ( A. x  e.  A  ( card `  B )  =  B  ->  U_ x  e.  A  ( card `  B
)  =  U_ x  e.  A  B )
2120adantl 452 . . . 4  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  U_ x  e.  A  ( card `  B )  =  U_ x  e.  A  B
)
2221fveq2d 5529 . . 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 2323 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  (
card `  B )  =  B )  ->  ( card `  U_ x  e.  A  B )  = 
U_ x  e.  A  B )
2423ex 423 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   U.cuni 3827   U_ciun 3905   ` cfv 5255   cardccrd 7568
This theorem is referenced by:  alephcard  7697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572
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