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Theorem cardiun 7631
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 5780 . . . . . 6  |-  ( A  e.  V  ->  { z  |  E. x  e.  A  z  =  (
card `  B ) }  e.  _V )
2 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2302 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  ( card `  B )  <->  y  =  ( card `  B )
) )
43rexbidv 2577 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. x  e.  A  z  =  ( card `  B )  <->  E. x  e.  A  y  =  ( card `  B )
) )
52, 4elab 2927 . . . . . . . 8  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  <->  E. x  e.  A  y  =  ( card `  B )
)
6 cardidm 7608 . . . . . . . . . 10  |-  ( card `  ( card `  B
) )  =  (
card `  B )
7 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  (
card `  ( card `  B ) ) )
8 id 19 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  y  =  ( card `  B )
)
96, 7, 83eqtr4a 2354 . . . . . . . . 9  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  y )
109rexlimivw 2676 . . . . . . . 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 2621 . . . . . 6  |-  A. y  e.  { z  |  E. x  e.  A  z  =  ( card `  B
) }  ( card `  y )  =  y
13 carduni 7630 . . . . . 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 5555 . . . . . . 7  |-  ( card `  B )  e.  _V
1615dfiun2 3953 . . . . . 6  |-  U_ x  e.  A  ( card `  B )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) }
1716fveq2i 5544 . . . . 5  |-  ( card `  U_ x  e.  A  ( card `  B )
)  =  ( card `  U. { z  |  E. x  e.  A  z  =  ( card `  B ) } )
1814, 17, 163eqtr4g 2353 . . . 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 3937 . . . . 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 5545 . . 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 2336 . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   U.cuni 3843   U_ciun 3921   ` cfv 5271   cardccrd 7584
This theorem is referenced by:  alephcard  7713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-card 7588
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