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

Theorem inresflem 6913
Description: Lemma for inlresf1 6914 and inrresf1 6915. (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
inresflem.1  |-  F : A
-1-1-onto-> ( { X }  X.  A )
inresflem.2  |-  ( x  e.  A  ->  ( F `  x )  e.  B )
Assertion
Ref Expression
inresflem  |-  F : A -1-1-> B
Distinct variable groups:    x, A    x, B    x, F
Allowed substitution hint:    X( x)

Proof of Theorem inresflem
StepHypRef Expression
1 inresflem.1 . . 3  |-  F : A
-1-1-onto-> ( { X }  X.  A )
2 f1of1 5334 . . 3  |-  ( F : A -1-1-onto-> ( { X }  X.  A )  ->  F : A -1-1-> ( { X }  X.  A ) )
31, 2ax-mp 5 . 2  |-  F : A -1-1-> ( { X }  X.  A )
4 f1ofn 5336 . . . 4  |-  ( F : A -1-1-onto-> ( { X }  X.  A )  ->  F  Fn  A )
51, 4ax-mp 5 . . 3  |-  F  Fn  A
6 inresflem.2 . . . . 5  |-  ( x  e.  A  ->  ( F `  x )  e.  B )
76rgen 2462 . . . 4  |-  A. x  e.  A  ( F `  x )  e.  B
8 fnfvrnss 5548 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
95, 7, 8mp2an 422 . . 3  |-  ran  F  C_  B
10 df-f 5097 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
115, 9, 10mpbir2an 911 . 2  |-  F : A
--> B
12 f1ff1 5306 . 2  |-  ( ( F : A -1-1-> ( { X }  X.  A )  /\  F : A --> B )  ->  F : A -1-1-> B )
133, 11, 12mp2an 422 1  |-  F : A -1-1-> B
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   A.wral 2393    C_ wss 3041   {csn 3497    X. cxp 4507   ran crn 4510    Fn wfn 5088   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-f1o 5100  df-fv 5101
This theorem is referenced by:  inlresf1  6914  inrresf1  6915
  Copyright terms: Public domain W3C validator