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Theorem r111 7997
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 7989 . . 3  |-  R1  Fn  On
2 dffn2 5575 . . 3  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
31, 2mpbi 208 . 2  |-  R1 : On
--> _V
4 eloni 4744 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
5 eloni 4744 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
6 ordtri3or 4766 . . . . 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 7463 . . . . . . . . 9  |-  -.  ( R1 `  y )  ~< 
( R1 `  y
)
9 r1sdom 7996 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( R1 `  x
)  ~<  ( R1 `  y ) )
10 breq1 4310 . . . . . . . . . 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 1006 . . . . . . 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 1189 . . . . 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 7996 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
19 breq2 4311 . . . . . . . . . 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 1007 . . . . . . 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 1189 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) ) )
2515, 17, 243jaod 1282 . . . 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 2797 . 2  |-  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y )
28 dff13 5986 . 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 911 1  |-  R1 : On
-1-1-> _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987   class class class wbr 4307   Ord word 4733   Oncon0 4734    Fn wfn 5428   -->wf 5429   -1-1->wf1 5430   ` cfv 5433    ~< csdm 7324   R1cr1 7984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-r1 7986
This theorem is referenced by:  tskinf  8951  grothomex  9011  rankeq1o  28224  elhf  28227  hfninf  28239
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