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Theorem fin1a2lem2 8282
Description: Lemma for fin1a2 8296. (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 4794 . . 3  |-  ( x  e.  On  ->  suc  x  e.  On )
31, 2fmpti 5893 . 2  |-  S : On
--> On
41fin1a2lem1 8281 . . . . . 6  |-  ( a  e.  On  ->  ( S `  a )  =  suc  a )
51fin1a2lem1 8281 . . . . . 6  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
64, 5eqeqan12d 2452 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <->  suc  a  =  suc  b ) )
7 suc11 4686 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( suc  a  =  suc  b  <->  a  =  b ) )
86, 7bitrd 246 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <-> 
a  =  b ) )
98biimpd 200 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  ->  a  =  b ) )
109rgen2a 2773 . 2  |-  A. a  e.  On  A. b  e.  On  ( ( S `
 a )  =  ( S `  b
)  ->  a  =  b )
11 dff13 6005 . 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 888 1  |-  S : On
-1-1-> On
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706    e. cmpt 4267   Oncon0 4582   suc csuc 4584   -->wf 5451   -1-1->wf1 5452   ` cfv 5455
This theorem is referenced by:  fin1a2lem6  8286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fv 5463
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