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Theorem fin1a2lem2 8793
Description: Lemma for fin1a2 8807. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.a  |-  S  =  ( x  e.  On  |->  suc  x )
Assertion
Ref Expression
fin1a2lem2  |-  S : On
-1-1-> On

Proof of Theorem fin1a2lem2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin1a2lem.a . . 3  |-  S  =  ( x  e.  On  |->  suc  x )
2 suceloni 6643 . . 3  |-  ( x  e.  On  ->  suc  x  e.  On )
31, 2fmpti 6055 . 2  |-  S : On
--> On
41fin1a2lem1 8792 . . . . . 6  |-  ( a  e.  On  ->  ( S `  a )  =  suc  a )
51fin1a2lem1 8792 . . . . . 6  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
64, 5eqeqan12d 2490 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <->  suc  a  =  suc  b ) )
7 suc11 4987 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( suc  a  =  suc  b  <->  a  =  b ) )
86, 7bitrd 253 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <-> 
a  =  b ) )
98biimpd 207 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  ->  a  =  b ) )
109rgen2a 2894 . 2  |-  A. a  e.  On  A. b  e.  On  ( ( S `
 a )  =  ( S `  b
)  ->  a  =  b )
11 dff13 6165 . 2  |-  ( S : On -1-1-> On  <->  ( S : On --> On  /\  A. a  e.  On  A. b  e.  On  ( ( S `
 a )  =  ( S `  b
)  ->  a  =  b ) ) )
123, 10, 11mpbir2an 918 1  |-  S : On
-1-1-> On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    |-> cmpt 4511   Oncon0 4884   suc csuc 4886   -->wf 5590   -1-1->wf1 5591   ` cfv 5594
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-sep 4574  ax-nul 4582  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-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  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-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-fv 5602
This theorem is referenced by:  fin1a2lem6  8797
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