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Theorem rankuni 8352
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 4198 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
21fveq2d 5883 . . . 4  |-  ( x  =  A  ->  ( rank `  U. x )  =  ( rank `  U. A ) )
3 fveq2 5879 . . . . 5  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43unieqd 4200 . . . 4  |-  ( x  =  A  ->  U. ( rank `  x )  = 
U. ( rank `  A
) )
52, 4eqeq12d 2486 . . 3  |-  ( x  =  A  ->  (
( rank `  U. x )  =  U. ( rank `  x )  <->  ( rank ` 
U. A )  = 
U. ( rank `  A
) ) )
6 vex 3034 . . . . . . 7  |-  x  e. 
_V
76rankuni2 8344 . . . . . 6  |-  ( rank `  U. x )  = 
U_ z  e.  x  ( rank `  z )
8 fvex 5889 . . . . . . 7  |-  ( rank `  z )  e.  _V
98dfiun2 4303 . . . . . 6  |-  U_ z  e.  x  ( rank `  z )  =  U. { y  |  E. z  e.  x  y  =  ( rank `  z
) }
107, 9eqtri 2493 . . . . 5  |-  ( rank `  U. x )  = 
U. { y  |  E. z  e.  x  y  =  ( rank `  z ) }
11 df-rex 2762 . . . . . . . 8  |-  ( E. z  e.  x  y  =  ( rank `  z
)  <->  E. z ( z  e.  x  /\  y  =  ( rank `  z
) ) )
126rankel 8328 . . . . . . . . . . 11  |-  ( z  e.  x  ->  ( rank `  z )  e.  ( rank `  x
) )
1312anim1i 578 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  y  =  ( rank `  z ) )  -> 
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
1413eximi 1715 . . . . . . . . 9  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  E. z
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
15 19.42v 1842 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
16 eleq1 2537 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  z
)  ->  ( y  e.  ( rank `  x
)  <->  ( rank `  z
)  e.  ( rank `  x ) ) )
1716pm5.32ri 650 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  y  =  ( rank `  z
) )  <->  ( ( rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
1817exbii 1726 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  E. z ( (
rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
19 simpl 464 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  ->  y  e.  (
rank `  x )
)
20 rankon 8284 . . . . . . . . . . . . . . . . 17  |-  ( rank `  x )  e.  On
2120oneli 5537 . . . . . . . . . . . . . . . 16  |-  ( y  e.  ( rank `  x
)  ->  y  e.  On )
22 r1fnon 8256 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
23 fndm 5685 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
2521, 24syl6eleqr 2560 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( rank `  x
)  ->  y  e.  dom  R1 )
26 rankr1id 8351 . . . . . . . . . . . . . . 15  |-  ( y  e.  dom  R1  <->  ( rank `  ( R1 `  y
) )  =  y )
2725, 26sylib 201 . . . . . . . . . . . . . 14  |-  ( y  e.  ( rank `  x
)  ->  ( rank `  ( R1 `  y
) )  =  y )
2827eqcomd 2477 . . . . . . . . . . . . 13  |-  ( y  e.  ( rank `  x
)  ->  y  =  ( rank `  ( R1 `  y ) ) )
29 fvex 5889 . . . . . . . . . . . . . 14  |-  ( R1
`  y )  e. 
_V
30 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R1 `  y )  ->  ( rank `  z )  =  ( rank `  ( R1 `  y ) ) )
3130eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( z  =  ( R1 `  y )  ->  (
y  =  ( rank `  z )  <->  y  =  ( rank `  ( R1 `  y ) ) ) )
3229, 31spcev 3127 . . . . . . . . . . . . 13  |-  ( y  =  ( rank `  ( R1 `  y ) )  ->  E. z  y  =  ( rank `  z
) )
3328, 32syl 17 . . . . . . . . . . . 12  |-  ( y  e.  ( rank `  x
)  ->  E. z 
y  =  ( rank `  z ) )
3433ancli 560 . . . . . . . . . . 11  |-  ( y  e.  ( rank `  x
)  ->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
3519, 34impbii 192 . . . . . . . . . 10  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3615, 18, 353bitr3i 283 . . . . . . . . 9  |-  ( E. z ( ( rank `  z )  e.  (
rank `  x )  /\  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3714, 36sylib 201 . . . . . . . 8  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  y  e.  ( rank `  x
) )
3811, 37sylbi 200 . . . . . . 7  |-  ( E. z  e.  x  y  =  ( rank `  z
)  ->  y  e.  ( rank `  x )
)
3938abssi 3490 . . . . . 6  |-  { y  |  E. z  e.  x  y  =  (
rank `  z ) }  C_  ( rank `  x
)
4039unissi 4213 . . . . 5  |-  U. {
y  |  E. z  e.  x  y  =  ( rank `  z ) }  C_  U. ( rank `  x )
4110, 40eqsstri 3448 . . . 4  |-  ( rank `  U. x )  C_  U. ( rank `  x
)
42 pwuni 4631 . . . . . . . 8  |-  x  C_  ~P U. x
436uniex 6606 . . . . . . . . . 10  |-  U. x  e.  _V
4443pwex 4584 . . . . . . . . 9  |-  ~P U. x  e.  _V
4544rankss 8338 . . . . . . . 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 8332 . . . . . . 7  |-  ( rank `  ~P U. x )  =  suc  ( rank `  U. x )
4846, 47sseqtri 3450 . . . . . 6  |-  ( rank `  x )  C_  suc  ( rank `  U. x )
4948unissi 4213 . . . . 5  |-  U. ( rank `  x )  C_  U.
suc  ( rank `  U. x )
50 rankon 8284 . . . . . 6  |-  ( rank `  U. x )  e.  On
5150onunisuci 5543 . . . . 5  |-  U. suc  ( rank `  U. x )  =  ( rank `  U. x )
5249, 51sseqtri 3450 . . . 4  |-  U. ( rank `  x )  C_  ( rank `  U. x )
5341, 52eqssi 3434 . . 3  |-  ( rank `  U. x )  = 
U. ( rank `  x
)
545, 53vtoclg 3093 . 2  |-  ( A  e.  _V  ->  ( rank `  U. A )  =  U. ( rank `  A ) )
55 uniexb 6620 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
56 fvprc 5873 . . . . 5  |-  ( -. 
U. A  e.  _V  ->  ( rank `  U. A )  =  (/) )
5755, 56sylnbi 313 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  (/) )
58 uni0 4217 . . . 4  |-  U. (/)  =  (/)
5957, 58syl6eqr 2523 . . 3  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. (/) )
60 fvprc 5873 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
6160unieqd 4200 . . 3  |-  ( -.  A  e.  _V  ->  U. ( rank `  A
)  =  U. (/) )
6259, 61eqtr4d 2508 . 2  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. ( rank `  A ) )
6354, 62pm2.61i 169 1  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   U_ciun 4269   dom cdm 4839   Oncon0 5430   suc csuc 5432    Fn wfn 5584   ` cfv 5589   R1cr1 8251   rankcrnk 8252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-r1 8253  df-rank 8254
This theorem is referenced by:  rankuniss  8355  rankbnd2  8358  rankxplim2  8369  rankxplim3  8370  rankxpsuc  8371  r1limwun  9179  hfuni  31022
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