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Theorem r111 8205
Description: The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
Assertion
Ref Expression
r111  |-  R1 : On
-1-1-> _V

Proof of Theorem r111
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1fnon 8197 . . 3  |-  R1  Fn  On
2 dffn2 5738 . . 3  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
31, 2mpbi 208 . 2  |-  R1 : On
--> _V
4 eloni 4894 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
5 eloni 4894 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
6 ordtri3or 4916 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
74, 5, 6syl2an 477 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
8 sdomirr 7666 . . . . . . . . 9  |-  -.  ( R1 `  y )  ~< 
( R1 `  y
)
9 r1sdom 8204 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( R1 `  x
)  ~<  ( R1 `  y ) )
10 breq1 4456 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  x
)  ~<  ( R1 `  y )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
119, 10syl5ibcom 220 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
128, 11mtoi 178 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
13123adant1 1014 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
1413pm2.21d 106 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
15143expia 1198 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y ) ) )
16 ax-1 6 . . . . . 6  |-  ( x  =  y  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
1716a1i 11 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  =  y  ->  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y ) ) )
18 r1sdom 8204 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
19 breq2 4457 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  y
)  ~<  ( R1 `  x )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
2018, 19syl5ibcom 220 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
218, 20mtoi 178 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
22213adant2 1015 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
2322pm2.21d 106 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
24233expia 1198 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) ) )
2515, 17, 243jaod 1292 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
) )
267, 25mpd 15 . . 3  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) )
2726rgen2a 2894 . 2  |-  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y )
28 dff13 6165 . 2  |-  ( R1 : On -1-1-> _V  <->  ( R1 : On --> _V  /\  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y ) ) )
293, 27, 28mpbir2an 918 1  |-  R1 : On
-1-1-> _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   class class class wbr 4453   Ord word 4883   Oncon0 4884    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   ` cfv 5594    ~< csdm 7527   R1cr1 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-r1 8194
This theorem is referenced by:  tskinf  9159  grothomex  9219  rankeq1o  29755  elhf  29758  hfninf  29770
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