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Theorem fin1a2lem2 8027
Description: Lemma for fin1a2 8041. (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 4604 . . 3  |-  ( x  e.  On  ->  suc  x  e.  On )
31, 2fmpti 5683 . 2  |-  S : On
--> On
41fin1a2lem1 8026 . . . . . 6  |-  ( a  e.  On  ->  ( S `  a )  =  suc  a )
51fin1a2lem1 8026 . . . . . 6  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
64, 5eqeqan12d 2298 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <->  suc  a  =  suc  b ) )
7 suc11 4496 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( suc  a  =  suc  b  <->  a  =  b ) )
86, 7bitrd 244 . . . 4  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <-> 
a  =  b ) )
98biimpd 198 . . 3  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  ->  a  =  b ) )
109rgen2a 2609 . 2  |-  A. a  e.  On  A. b  e.  On  ( ( S `
 a )  =  ( S `  b
)  ->  a  =  b )
11 dff13 5783 . 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 886 1  |-  S : On
-1-1-> On
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   Oncon0 4392   suc csuc 4394   -->wf 5251   -1-1->wf1 5252   ` cfv 5255
This theorem is referenced by:  fin1a2lem6  8031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263
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