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Theorem r111 8184
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 8176 . . 3  |-  R1  Fn  On
2 dffn2 5714 . . 3  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
31, 2mpbi 208 . 2  |-  R1 : On
--> _V
4 eloni 4877 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
5 eloni 4877 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
6 ordtri3or 4899 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
74, 5, 6syl2an 475 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
8 sdomirr 7647 . . . . . . . . 9  |-  -.  ( R1 `  y )  ~< 
( R1 `  y
)
9 r1sdom 8183 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( R1 `  x
)  ~<  ( R1 `  y ) )
10 breq1 4442 . . . . . . . . . 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 1012 . . . . . . 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 1196 . . . . 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 8183 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
19 breq2 4443 . . . . . . . . . 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 1013 . . . . . . 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 1196 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) ) )
2515, 17, 243jaod 1290 . . . 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 2881 . 2  |-  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y )
28 dff13 6141 . 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 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   class class class wbr 4439   Ord word 4866   Oncon0 4867    Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   ` cfv 5570    ~< csdm 7508   R1cr1 8171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-r1 8173
This theorem is referenced by:  tskinf  9136  grothomex  9196  rankeq1o  30059  elhf  30062  hfninf  30074
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