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Theorem cardiun 8362
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 6759 . . . . . 6  |-  ( A  e.  V  ->  { z  |  E. x  e.  A  z  =  (
card `  B ) }  e.  _V )
2 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2471 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  ( card `  B )  <->  y  =  ( card `  B )
) )
43rexbidv 2973 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. x  e.  A  z  =  ( card `  B )  <->  E. x  e.  A  y  =  ( card `  B )
) )
52, 4elab 3250 . . . . . . . 8  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  <->  E. x  e.  A  y  =  ( card `  B )
)
6 cardidm 8339 . . . . . . . . . 10  |-  ( card `  ( card `  B
) )  =  (
card `  B )
7 fveq2 5865 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  (
card `  ( card `  B ) ) )
8 id 22 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  y  =  ( card `  B )
)
96, 7, 83eqtr4a 2534 . . . . . . . . 9  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  y )
109rexlimivw 2952 . . . . . . . 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 2824 . . . . . 6  |-  A. y  e.  { z  |  E. x  e.  A  z  =  ( card `  B
) }  ( card `  y )  =  y
13 carduni 8361 . . . . . 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 5875 . . . . . . 7  |-  ( card `  B )  e.  _V
1615dfiun2 4359 . . . . . 6  |-  U_ x  e.  A  ( card `  B )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) }
1716fveq2i 5868 . . . . 5  |-  ( card `  U_ x  e.  A  ( card `  B )
)  =  ( card `  U. { z  |  E. x  e.  A  z  =  ( card `  B ) } )
1814, 17, 163eqtr4g 2533 . . . 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 4342 . . . . 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 5869 . . 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 2516 . 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113   U.cuni 4245   U_ciun 4325   ` cfv 5587   cardccrd 8315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-card 8319
This theorem is referenced by:  alephcard  8450
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