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Theorem rankung 24206
Description: The rank of the union of two sets. Closed form of rankun 7524. (Contributed by Scott Fenton, 15-Jul-2015.)
Assertion
Ref Expression
rankung  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( rank `  ( A  u.  B )
)  =  ( (
rank `  A )  u.  ( rank `  B
) ) )

Proof of Theorem rankung
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3323 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21fveq2d 5490 . . 3  |-  ( x  =  A  ->  ( rank `  ( x  u.  y ) )  =  ( rank `  ( A  u.  y )
) )
3 fveq2 5486 . . . 4  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43uneq1d 3329 . . 3  |-  ( x  =  A  ->  (
( rank `  x )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) )
52, 4eqeq12d 2298 . 2  |-  ( x  =  A  ->  (
( rank `  ( x  u.  y ) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) ) )
6 uneq2 3324 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
76fveq2d 5490 . . 3  |-  ( y  =  B  ->  ( rank `  ( A  u.  y ) )  =  ( rank `  ( A  u.  B )
) )
8 fveq2 5486 . . . 4  |-  ( y  =  B  ->  ( rank `  y )  =  ( rank `  B
) )
98uneq2d 3330 . . 3  |-  ( y  =  B  ->  (
( rank `  A )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) )
107, 9eqeq12d 2298 . 2  |-  ( y  =  B  ->  (
( rank `  ( A  u.  y ) )  =  ( ( rank `  A
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) ) )
11 vex 2792 . . 3  |-  x  e. 
_V
12 vex 2792 . . 3  |-  y  e. 
_V
1311, 12rankun 7524 . 2  |-  ( rank `  ( x  u.  y
) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )
145, 10, 13vtocl2g 2848 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( rank `  ( A  u.  B )
)  =  ( (
rank `  A )  u.  ( rank `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685    u. cun 3151   ` cfv 5221   rankcrnk 7431
This theorem is referenced by:  hfun  24218
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7302  ax-inf2 7338
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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