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Theorem cardiun 8164
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 6564 . . . . . 6  |-  ( A  e.  V  ->  { z  |  E. x  e.  A  z  =  (
card `  B ) }  e.  _V )
2 vex 2987 . . . . . . . . 9  |-  y  e. 
_V
3 eqeq1 2449 . . . . . . . . . 10  |-  ( z  =  y  ->  (
z  =  ( card `  B )  <->  y  =  ( card `  B )
) )
43rexbidv 2748 . . . . . . . . 9  |-  ( z  =  y  ->  ( E. x  e.  A  z  =  ( card `  B )  <->  E. x  e.  A  y  =  ( card `  B )
) )
52, 4elab 3118 . . . . . . . 8  |-  ( y  e.  { z  |  E. x  e.  A  z  =  ( card `  B ) }  <->  E. x  e.  A  y  =  ( card `  B )
)
6 cardidm 8141 . . . . . . . . . 10  |-  ( card `  ( card `  B
) )  =  (
card `  B )
7 fveq2 5703 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  (
card `  ( card `  B ) ) )
8 id 22 . . . . . . . . . 10  |-  ( y  =  ( card `  B
)  ->  y  =  ( card `  B )
)
96, 7, 83eqtr4a 2501 . . . . . . . . 9  |-  ( y  =  ( card `  B
)  ->  ( card `  y )  =  y )
109rexlimivw 2849 . . . . . . . 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 2793 . . . . . 6  |-  A. y  e.  { z  |  E. x  e.  A  z  =  ( card `  B
) }  ( card `  y )  =  y
13 carduni 8163 . . . . . 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 5713 . . . . . . 7  |-  ( card `  B )  e.  _V
1615dfiun2 4216 . . . . . 6  |-  U_ x  e.  A  ( card `  B )  =  U. { z  |  E. x  e.  A  z  =  ( card `  B
) }
1716fveq2i 5706 . . . . 5  |-  ( card `  U_ x  e.  A  ( card `  B )
)  =  ( card `  U. { z  |  E. x  e.  A  z  =  ( card `  B ) } )
1814, 17, 163eqtr4g 2500 . . . 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 4199 . . . . 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 5707 . . 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 2483 . 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 1369    e. wcel 1756   {cab 2429   A.wral 2727   E.wrex 2728   _Vcvv 2984   U.cuni 4103   U_ciun 4183   ` cfv 5430   cardccrd 8117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-card 8121
This theorem is referenced by:  alephcard  8252
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