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Theorem rankuni 8334
Description: The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankuni  |-  ( rank `  U. A )  = 
U. ( rank `  A
)

Proof of Theorem rankuni
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4206 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
21fveq2d 5869 . . . 4  |-  ( x  =  A  ->  ( rank `  U. x )  =  ( rank `  U. A ) )
3 fveq2 5865 . . . . 5  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43unieqd 4208 . . . 4  |-  ( x  =  A  ->  U. ( rank `  x )  = 
U. ( rank `  A
) )
52, 4eqeq12d 2466 . . 3  |-  ( x  =  A  ->  (
( rank `  U. x )  =  U. ( rank `  x )  <->  ( rank ` 
U. A )  = 
U. ( rank `  A
) ) )
6 vex 3048 . . . . . . 7  |-  x  e. 
_V
76rankuni2 8326 . . . . . 6  |-  ( rank `  U. x )  = 
U_ z  e.  x  ( rank `  z )
8 fvex 5875 . . . . . . 7  |-  ( rank `  z )  e.  _V
98dfiun2 4312 . . . . . 6  |-  U_ z  e.  x  ( rank `  z )  =  U. { y  |  E. z  e.  x  y  =  ( rank `  z
) }
107, 9eqtri 2473 . . . . 5  |-  ( rank `  U. x )  = 
U. { y  |  E. z  e.  x  y  =  ( rank `  z ) }
11 df-rex 2743 . . . . . . . 8  |-  ( E. z  e.  x  y  =  ( rank `  z
)  <->  E. z ( z  e.  x  /\  y  =  ( rank `  z
) ) )
126rankel 8310 . . . . . . . . . . 11  |-  ( z  e.  x  ->  ( rank `  z )  e.  ( rank `  x
) )
1312anim1i 572 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  y  =  ( rank `  z ) )  -> 
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
1413eximi 1707 . . . . . . . . 9  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  E. z
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
15 19.42v 1834 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
16 eleq1 2517 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  z
)  ->  ( y  e.  ( rank `  x
)  <->  ( rank `  z
)  e.  ( rank `  x ) ) )
1716pm5.32ri 644 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  y  =  ( rank `  z
) )  <->  ( ( rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
1817exbii 1718 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  E. z ( (
rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
19 simpl 459 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  ->  y  e.  (
rank `  x )
)
20 rankon 8266 . . . . . . . . . . . . . . . . 17  |-  ( rank `  x )  e.  On
2120oneli 5530 . . . . . . . . . . . . . . . 16  |-  ( y  e.  ( rank `  x
)  ->  y  e.  On )
22 r1fnon 8238 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
23 fndm 5675 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
2521, 24syl6eleqr 2540 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( rank `  x
)  ->  y  e.  dom  R1 )
26 rankr1id 8333 . . . . . . . . . . . . . . 15  |-  ( y  e.  dom  R1  <->  ( rank `  ( R1 `  y
) )  =  y )
2725, 26sylib 200 . . . . . . . . . . . . . 14  |-  ( y  e.  ( rank `  x
)  ->  ( rank `  ( R1 `  y
) )  =  y )
2827eqcomd 2457 . . . . . . . . . . . . 13  |-  ( y  e.  ( rank `  x
)  ->  y  =  ( rank `  ( R1 `  y ) ) )
29 fvex 5875 . . . . . . . . . . . . . 14  |-  ( R1
`  y )  e. 
_V
30 fveq2 5865 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R1 `  y )  ->  ( rank `  z )  =  ( rank `  ( R1 `  y ) ) )
3130eqeq2d 2461 . . . . . . . . . . . . . 14  |-  ( z  =  ( R1 `  y )  ->  (
y  =  ( rank `  z )  <->  y  =  ( rank `  ( R1 `  y ) ) ) )
3229, 31spcev 3141 . . . . . . . . . . . . 13  |-  ( y  =  ( rank `  ( R1 `  y ) )  ->  E. z  y  =  ( rank `  z
) )
3328, 32syl 17 . . . . . . . . . . . 12  |-  ( y  e.  ( rank `  x
)  ->  E. z 
y  =  ( rank `  z ) )
3433ancli 554 . . . . . . . . . . 11  |-  ( y  e.  ( rank `  x
)  ->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
3519, 34impbii 191 . . . . . . . . . 10  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3615, 18, 353bitr3i 279 . . . . . . . . 9  |-  ( E. z ( ( rank `  z )  e.  (
rank `  x )  /\  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3714, 36sylib 200 . . . . . . . 8  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  y  e.  ( rank `  x
) )
3811, 37sylbi 199 . . . . . . 7  |-  ( E. z  e.  x  y  =  ( rank `  z
)  ->  y  e.  ( rank `  x )
)
3938abssi 3504 . . . . . 6  |-  { y  |  E. z  e.  x  y  =  (
rank `  z ) }  C_  ( rank `  x
)
4039unissi 4221 . . . . 5  |-  U. {
y  |  E. z  e.  x  y  =  ( rank `  z ) }  C_  U. ( rank `  x )
4110, 40eqsstri 3462 . . . 4  |-  ( rank `  U. x )  C_  U. ( rank `  x
)
42 pwuni 4631 . . . . . . . 8  |-  x  C_  ~P U. x
436uniex 6587 . . . . . . . . . 10  |-  U. x  e.  _V
4443pwex 4586 . . . . . . . . 9  |-  ~P U. x  e.  _V
4544rankss 8320 . . . . . . . 8  |-  ( x 
C_  ~P U. x  -> 
( rank `  x )  C_  ( rank `  ~P U. x ) )
4642, 45ax-mp 5 . . . . . . 7  |-  ( rank `  x )  C_  ( rank `  ~P U. x
)
4743rankpw 8314 . . . . . . 7  |-  ( rank `  ~P U. x )  =  suc  ( rank `  U. x )
4846, 47sseqtri 3464 . . . . . 6  |-  ( rank `  x )  C_  suc  ( rank `  U. x )
4948unissi 4221 . . . . 5  |-  U. ( rank `  x )  C_  U.
suc  ( rank `  U. x )
50 rankon 8266 . . . . . 6  |-  ( rank `  U. x )  e.  On
5150onunisuci 5536 . . . . 5  |-  U. suc  ( rank `  U. x )  =  ( rank `  U. x )
5249, 51sseqtri 3464 . . . 4  |-  U. ( rank `  x )  C_  ( rank `  U. x )
5341, 52eqssi 3448 . . 3  |-  ( rank `  U. x )  = 
U. ( rank `  x
)
545, 53vtoclg 3107 . 2  |-  ( A  e.  _V  ->  ( rank `  U. A )  =  U. ( rank `  A ) )
55 uniexb 6601 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
56 fvprc 5859 . . . . 5  |-  ( -. 
U. A  e.  _V  ->  ( rank `  U. A )  =  (/) )
5755, 56sylnbi 308 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  (/) )
58 uni0 4225 . . . 4  |-  U. (/)  =  (/)
5957, 58syl6eqr 2503 . . 3  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. (/) )
60 fvprc 5859 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
6160unieqd 4208 . . 3  |-  ( -.  A  e.  _V  ->  U. ( rank `  A
)  =  U. (/) )
6259, 61eqtr4d 2488 . 2  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. ( rank `  A ) )
6354, 62pm2.61i 168 1  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   U.cuni 4198   U_ciun 4278   dom cdm 4834   Oncon0 5423   suc csuc 5425    Fn wfn 5577   ` cfv 5582   R1cr1 8233   rankcrnk 8234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-r1 8235  df-rank 8236
This theorem is referenced by:  rankuniss  8337  rankbnd2  8340  rankxplim2  8351  rankxplim3  8352  rankxpsuc  8353  r1limwun  9161  hfuni  30951
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