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Theorem r111 7661
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 7653 . . 3  |-  R1  Fn  On
2 dffn2 5555 . . 3  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
31, 2mpbi 200 . 2  |-  R1 : On
--> _V
4 eloni 4555 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
5 eloni 4555 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
6 ordtri3or 4577 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
74, 5, 6syl2an 464 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
8 sdomirr 7207 . . . . . . . . 9  |-  -.  ( R1 `  y )  ~< 
( R1 `  y
)
9 r1sdom 7660 . . . . . . . . . 10  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( R1 `  x
)  ~<  ( R1 `  y ) )
10 breq1 4179 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  x
)  ~<  ( R1 `  y )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
119, 10syl5ibcom 212 . . . . . . . . 9  |-  ( ( y  e.  On  /\  x  e.  y )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
128, 11mtoi 171 . . . . . . . 8  |-  ( ( y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
13123adant1 975 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
1413pm2.21d 100 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  x  e.  y )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
15143expia 1155 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  e.  y  ->  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y ) ) )
16 ax-1 5 . . . . . 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 7660 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
19 breq2 4180 . . . . . . . . . 10  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  (
( R1 `  y
)  ~<  ( R1 `  x )  <->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
2018, 19syl5ibcom 212 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( R1 `  y )  ~<  ( R1 `  y ) ) )
218, 20mtoi 171 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
22213adant2 976 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  -.  ( R1 `  x )  =  ( R1 `  y ) )
2322pm2.21d 100 . . . . . 6  |-  ( ( x  e.  On  /\  y  e.  On  /\  y  e.  x )  ->  (
( R1 `  x
)  =  ( R1
`  y )  ->  x  =  y )
)
24233expia 1155 . . . . 5  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( y  e.  x  ->  ( ( R1 `  x )  =  ( R1 `  y )  ->  x  =  y ) ) )
2515, 17, 243jaod 1248 . . . 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 2736 . 2  |-  A. x  e.  On  A. y  e.  On  ( ( R1
`  x )  =  ( R1 `  y
)  ->  x  =  y )
28 dff13 5967 . 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 887 1  |-  R1 : On
-1-1-> _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   class class class wbr 4176   Ord word 4544   Oncon0 4545    Fn wfn 5412   -->wf 5413   -1-1->wf1 5414   ` cfv 5417    ~< csdm 7071   R1cr1 7648
This theorem is referenced by:  tskinf  8604  grothomex  8664  rankeq1o  26020  elhf  26023  hfninf  26035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-r1 7650
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