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Theorem rankung 26011
Description: The rank of the union of two sets. Closed form of rankun 7738. (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 3454 . . . 4  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
21fveq2d 5691 . . 3  |-  ( x  =  A  ->  ( rank `  ( x  u.  y ) )  =  ( rank `  ( A  u.  y )
) )
3 fveq2 5687 . . . 4  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43uneq1d 3460 . . 3  |-  ( x  =  A  ->  (
( rank `  x )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) )
52, 4eqeq12d 2418 . 2  |-  ( x  =  A  ->  (
( rank `  ( x  u.  y ) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  y
) )  =  ( ( rank `  A
)  u.  ( rank `  y ) ) ) )
6 uneq2 3455 . . . 4  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
76fveq2d 5691 . . 3  |-  ( y  =  B  ->  ( rank `  ( A  u.  y ) )  =  ( rank `  ( A  u.  B )
) )
8 fveq2 5687 . . . 4  |-  ( y  =  B  ->  ( rank `  y )  =  ( rank `  B
) )
98uneq2d 3461 . . 3  |-  ( y  =  B  ->  (
( rank `  A )  u.  ( rank `  y
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) )
107, 9eqeq12d 2418 . 2  |-  ( y  =  B  ->  (
( rank `  ( A  u.  y ) )  =  ( ( rank `  A
)  u.  ( rank `  y ) )  <->  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) ) ) )
11 vex 2919 . . 3  |-  x  e. 
_V
12 vex 2919 . . 3  |-  y  e. 
_V
1311, 12rankun 7738 . 2  |-  ( rank `  ( x  u.  y
) )  =  ( ( rank `  x
)  u.  ( rank `  y ) )
145, 10, 13vtocl2g 2975 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 359    = wceq 1649    e. wcel 1721    u. cun 3278   ` cfv 5413   rankcrnk 7645
This theorem is referenced by:  hfun  26023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647
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