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

Theorem inresflem 7004
Description: Lemma for inlresf1 7005 and inrresf1 7006. (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 5413 . . 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 5415 . . . 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 2510 . . . 4  |-  A. x  e.  A  ( F `  x )  e.  B
8 fnfvrnss 5627 . . . 4  |-  ( ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  ->  ran  F  C_  B )
95, 7, 8mp2an 423 . . 3  |-  ran  F  C_  B
10 df-f 5174 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
115, 9, 10mpbir2an 927 . 2  |-  F : A
--> B
12 f1ff1 5383 . 2  |-  ( ( F : A -1-1-> ( { X }  X.  A )  /\  F : A --> B )  ->  F : A -1-1-> B )
133, 11, 12mp2an 423 1  |-  F : A -1-1-> B
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   A.wral 2435    C_ wss 3102   {csn 3560    X. cxp 4584   ran crn 4587    Fn wfn 5165   -->wf 5166   -1-1->wf1 5167   -1-1-onto->wf1o 5169   ` cfv 5170
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-f1o 5177  df-fv 5178
This theorem is referenced by:  inlresf1  7005  inrresf1  7006
  Copyright terms: Public domain W3C validator