ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1eq3 Unicode version

Theorem f1eq3 5117
Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1eq3  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )

Proof of Theorem f1eq3
StepHypRef Expression
1 feq3 5060 . . 3  |-  ( A  =  B  ->  ( F : C --> A  <->  F : C
--> B ) )
21anbi1d 446 . 2  |-  ( A  =  B  ->  (
( F : C --> A  /\  Fun  `' F
)  <->  ( F : C
--> B  /\  Fun  `' F ) ) )
3 df-f1 4935 . 2  |-  ( F : C -1-1-> A  <->  ( F : C --> A  /\  Fun  `' F ) )
4 df-f1 4935 . 2  |-  ( F : C -1-1-> B  <->  ( F : C --> B  /\  Fun  `' F ) )
52, 3, 43bitr4g 216 1  |-  ( A  =  B  ->  ( F : C -1-1-> A  <->  F : C -1-1-> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   `'ccnv 4372   Fun wfun 4924   -->wf 4926   -1-1->wf1 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-f 4934  df-f1 4935
This theorem is referenced by:  f1oeq3  5147  f1eq123d  5149  tposf12  5915  brdomg  6260
  Copyright terms: Public domain W3C validator