MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin1a2lem2 Structured version   Unicode version

Theorem fin1a2lem2 8582
Description: Lemma for fin1a2 8596. (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 6436 . . 3  |-  ( x  e.  On  ->  suc  x  e.  On )
31, 2fmpti 5878 . 2  |-  S : On
--> On
41fin1a2lem1 8581 . . . . . 6  |-  ( a  e.  On  ->  ( S `  a )  =  suc  a )
51fin1a2lem1 8581 . . . . . 6  |-  ( b  e.  On  ->  ( S `  b )  =  suc  b )
64, 5eqeqan12d 2458 . . . . 5  |-  ( ( a  e.  On  /\  b  e.  On )  ->  ( ( S `  a )  =  ( S `  b )  <->  suc  a  =  suc  b ) )
7 suc11 4834 . . . . 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 2794 . 2  |-  A. a  e.  On  A. b  e.  On  ( ( S `
 a )  =  ( S `  b
)  ->  a  =  b )
11 dff13 5983 . 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 911 1  |-  S : On
-1-1-> On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    e. cmpt 4362   Oncon0 4731   suc csuc 4733   -->wf 5426   -1-1->wf1 5427   ` cfv 5430
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-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
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 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fv 5438
This theorem is referenced by:  fin1a2lem6  8586
  Copyright terms: Public domain W3C validator