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Theorem rankung 24868
Description: The rank of the union of two sets. Closed form of rankun 7544. (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 3335 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( rank `  ( x  u.  y ) )  =  ( rank `  ( A  u.  y )
) )
3 fveq2 5541 . . . 4  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43uneq1d 3341 . . 3  |-  ( x  =  A  ->  (
( rank `  x )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) )
52, 4eqeq12d 2310 . 2  |-  ( x  =  A  ->  (
( rank `  ( x  u.  y ) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) ) )
6 uneq2 3336 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
76fveq2d 5545 . . 3  |-  ( y  =  B  ->  ( rank `  ( A  u.  y ) )  =  ( rank `  ( A  u.  B )
) )
8 fveq2 5541 . . . 4  |-  ( y  =  B  ->  ( rank `  y )  =  ( rank `  B
) )
98uneq2d 3342 . . 3  |-  ( y  =  B  ->  (
( rank `  A )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) )
107, 9eqeq12d 2310 . 2  |-  ( y  =  B  ->  (
( rank `  ( A  u.  y ) )  =  ( ( rank `  A
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) ) )
11 vex 2804 . . 3  |-  x  e. 
_V
12 vex 2804 . . 3  |-  y  e. 
_V
1311, 12rankun 7544 . 2  |-  ( rank `  ( x  u.  y
) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )
145, 10, 13vtocl2g 2860 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 1632    e. wcel 1696    u. cun 3163   ` cfv 5271   rankcrnk 7451
This theorem is referenced by:  hfun  24880
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  ax-reg 7322  ax-inf2 7358
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-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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