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Theorem r111 7704
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 7696 . . 3  |-  R1  Fn  On
2 dffn2 5595 . . 3  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
31, 2mpbi 201 . 2  |-  R1 : On
--> _V
4 eloni 4594 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
5 eloni 4594 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
6 ordtri3or 4616 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
74, 5, 6syl2an 465 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
8 sdomirr 7247 . . . . . . . . 9  |-  -.  ( R1 `  y )  ~< 
( R1 `  y
)
9 r1sdom 7703 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( R1 `  x
)  ~<  ( R1 `  y ) )
10 breq1 4218 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  x
)  ~<  ( R1 `  y )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
119, 10syl5ibcom 213 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
128, 11mtoi 172 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
13123adant1 976 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
1413pm2.21d 101 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
15143expia 1156 . . . . 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 7703 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
19 breq2 4219 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  y
)  ~<  ( R1 `  x )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
2018, 19syl5ibcom 213 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
218, 20mtoi 172 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
22213adant2 977 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
2322pm2.21d 101 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
24233expia 1156 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) ) )
2515, 17, 243jaod 1249 . . . 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 2774 . 2  |-  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y )
28 dff13 6007 . 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 888 1  |-  R1 : On
-1-1-> _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    \/ w3o 936    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   class class class wbr 4215   Ord word 4583   Oncon0 4584    Fn wfn 5452   -->wf 5453   -1-1->wf1 5454   ` cfv 5457    ~< csdm 7111   R1cr1 7691
This theorem is referenced by:  tskinf  8649  grothomex  8709  rankeq1o  26117  elhf  26120  hfninf  26132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-r1 7693
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