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Theorem rankuni 8324
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 4221 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
21fveq2d 5876 . . . 4  |-  ( x  =  A  ->  ( rank `  U. x )  =  ( rank `  U. A ) )
3 fveq2 5872 . . . . 5  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
43unieqd 4223 . . . 4  |-  ( x  =  A  ->  U. ( rank `  x )  = 
U. ( rank `  A
) )
52, 4eqeq12d 2442 . . 3  |-  ( x  =  A  ->  (
( rank `  U. x )  =  U. ( rank `  x )  <->  ( rank ` 
U. A )  = 
U. ( rank `  A
) ) )
6 vex 3081 . . . . . . 7  |-  x  e. 
_V
76rankuni2 8316 . . . . . 6  |-  ( rank `  U. x )  = 
U_ z  e.  x  ( rank `  z )
8 fvex 5882 . . . . . . 7  |-  ( rank `  z )  e.  _V
98dfiun2 4327 . . . . . 6  |-  U_ z  e.  x  ( rank `  z )  =  U. { y  |  E. z  e.  x  y  =  ( rank `  z
) }
107, 9eqtri 2449 . . . . 5  |-  ( rank `  U. x )  = 
U. { y  |  E. z  e.  x  y  =  ( rank `  z ) }
11 df-rex 2779 . . . . . . . 8  |-  ( E. z  e.  x  y  =  ( rank `  z
)  <->  E. z ( z  e.  x  /\  y  =  ( rank `  z
) ) )
126rankel 8300 . . . . . . . . . . 11  |-  ( z  e.  x  ->  ( rank `  z )  e.  ( rank `  x
) )
1312anim1i 570 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  y  =  ( rank `  z ) )  -> 
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
1413eximi 1702 . . . . . . . . 9  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  E. z
( ( rank `  z
)  e.  ( rank `  x )  /\  y  =  ( rank `  z
) ) )
15 19.42v 1823 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
16 eleq1 2492 . . . . . . . . . . . 12  |-  ( y  =  ( rank `  z
)  ->  ( y  e.  ( rank `  x
)  <->  ( rank `  z
)  e.  ( rank `  x ) ) )
1716pm5.32ri 642 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  y  =  ( rank `  z
) )  <->  ( ( rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
1817exbii 1712 . . . . . . . . . 10  |-  ( E. z ( y  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
)  <->  E. z ( (
rank `  z )  e.  ( rank `  x
)  /\  y  =  ( rank `  z )
) )
19 simpl 458 . . . . . . . . . . 11  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  ->  y  e.  (
rank `  x )
)
20 rankon 8256 . . . . . . . . . . . . . . . . 17  |-  ( rank `  x )  e.  On
2120oneli 5540 . . . . . . . . . . . . . . . 16  |-  ( y  e.  ( rank `  x
)  ->  y  e.  On )
22 r1fnon 8228 . . . . . . . . . . . . . . . . 17  |-  R1  Fn  On
23 fndm 5684 . . . . . . . . . . . . . . . . 17  |-  ( R1  Fn  On  ->  dom  R1  =  On )
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16  |-  dom  R1  =  On
2521, 24syl6eleqr 2519 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( rank `  x
)  ->  y  e.  dom  R1 )
26 rankr1id 8323 . . . . . . . . . . . . . . 15  |-  ( y  e.  dom  R1  <->  ( rank `  ( R1 `  y
) )  =  y )
2725, 26sylib 199 . . . . . . . . . . . . . 14  |-  ( y  e.  ( rank `  x
)  ->  ( rank `  ( R1 `  y
) )  =  y )
2827eqcomd 2428 . . . . . . . . . . . . 13  |-  ( y  e.  ( rank `  x
)  ->  y  =  ( rank `  ( R1 `  y ) ) )
29 fvex 5882 . . . . . . . . . . . . . 14  |-  ( R1
`  y )  e. 
_V
30 fveq2 5872 . . . . . . . . . . . . . . 15  |-  ( z  =  ( R1 `  y )  ->  ( rank `  z )  =  ( rank `  ( R1 `  y ) ) )
3130eqeq2d 2434 . . . . . . . . . . . . . 14  |-  ( z  =  ( R1 `  y )  ->  (
y  =  ( rank `  z )  <->  y  =  ( rank `  ( R1 `  y ) ) ) )
3229, 31spcev 3170 . . . . . . . . . . . . 13  |-  ( y  =  ( rank `  ( R1 `  y ) )  ->  E. z  y  =  ( rank `  z
) )
3328, 32syl 17 . . . . . . . . . . . 12  |-  ( y  e.  ( rank `  x
)  ->  E. z 
y  =  ( rank `  z ) )
3433ancli 553 . . . . . . . . . . 11  |-  ( y  e.  ( rank `  x
)  ->  ( y  e.  ( rank `  x
)  /\  E. z 
y  =  ( rank `  z ) ) )
3519, 34impbii 190 . . . . . . . . . 10  |-  ( ( y  e.  ( rank `  x )  /\  E. z  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3615, 18, 353bitr3i 278 . . . . . . . . 9  |-  ( E. z ( ( rank `  z )  e.  (
rank `  x )  /\  y  =  ( rank `  z ) )  <-> 
y  e.  ( rank `  x ) )
3714, 36sylib 199 . . . . . . . 8  |-  ( E. z ( z  e.  x  /\  y  =  ( rank `  z
) )  ->  y  e.  ( rank `  x
) )
3811, 37sylbi 198 . . . . . . 7  |-  ( E. z  e.  x  y  =  ( rank `  z
)  ->  y  e.  ( rank `  x )
)
3938abssi 3533 . . . . . 6  |-  { y  |  E. z  e.  x  y  =  (
rank `  z ) }  C_  ( rank `  x
)
4039unissi 4236 . . . . 5  |-  U. {
y  |  E. z  e.  x  y  =  ( rank `  z ) }  C_  U. ( rank `  x )
4110, 40eqsstri 3491 . . . 4  |-  ( rank `  U. x )  C_  U. ( rank `  x
)
42 pwuni 4644 . . . . . . . 8  |-  x  C_  ~P U. x
436uniex 6592 . . . . . . . . . 10  |-  U. x  e.  _V
4443pwex 4599 . . . . . . . . 9  |-  ~P U. x  e.  _V
4544rankss 8310 . . . . . . . 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 8304 . . . . . . 7  |-  ( rank `  ~P U. x )  =  suc  ( rank `  U. x )
4846, 47sseqtri 3493 . . . . . 6  |-  ( rank `  x )  C_  suc  ( rank `  U. x )
4948unissi 4236 . . . . 5  |-  U. ( rank `  x )  C_  U.
suc  ( rank `  U. x )
50 rankon 8256 . . . . . 6  |-  ( rank `  U. x )  e.  On
5150onunisuci 5546 . . . . 5  |-  U. suc  ( rank `  U. x )  =  ( rank `  U. x )
5249, 51sseqtri 3493 . . . 4  |-  U. ( rank `  x )  C_  ( rank `  U. x )
5341, 52eqssi 3477 . . 3  |-  ( rank `  U. x )  = 
U. ( rank `  x
)
545, 53vtoclg 3136 . 2  |-  ( A  e.  _V  ->  ( rank `  U. A )  =  U. ( rank `  A ) )
55 uniexb 6606 . . . . 5  |-  ( A  e.  _V  <->  U. A  e. 
_V )
56 fvprc 5866 . . . . 5  |-  ( -. 
U. A  e.  _V  ->  ( rank `  U. A )  =  (/) )
5755, 56sylnbi 307 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  (/) )
58 uni0 4240 . . . 4  |-  U. (/)  =  (/)
5957, 58syl6eqr 2479 . . 3  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. (/) )
60 fvprc 5866 . . . 4  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
6160unieqd 4223 . . 3  |-  ( -.  A  e.  _V  ->  U. ( rank `  A
)  =  U. (/) )
6259, 61eqtr4d 2464 . 2  |-  ( -.  A  e.  _V  ->  (
rank `  U. A )  =  U. ( rank `  A ) )
6354, 62pm2.61i 167 1  |-  ( rank `  U. A )  = 
U. ( rank `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405   E.wrex 2774   _Vcvv 3078    C_ wss 3433   (/)c0 3758   ~Pcpw 3976   U.cuni 4213   U_ciun 4293   dom cdm 4845   Oncon0 5433   suc csuc 5435    Fn wfn 5587   ` cfv 5592   R1cr1 8223   rankcrnk 8224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-reg 8098  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-r1 8225  df-rank 8226
This theorem is referenced by:  rankuniss  8327  rankbnd2  8330  rankxplim2  8341  rankxplim3  8342  rankxpsuc  8343  r1limwun  9150  hfuni  30733
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