Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1497 Structured version   Unicode version

Theorem bnj1497 33213
Description: Technical lemma for bnj60 33215. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1497.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1497.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1497.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
Assertion
Ref Expression
bnj1497  |-  A. g  e.  C  Fun  g
Distinct variable groups:    C, g    f, d    f, g
Allowed substitution hints:    A( x, f, g, d)    B( x, f, g, d)    C( x, f, d)    R( x, f, g, d)    G( x, f, g, d)    Y( x, f, g, d)

Proof of Theorem bnj1497
StepHypRef Expression
1 bnj1497.3 . . . . . 6  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
21bnj1317 32977 . . . . 5  |-  ( g  e.  C  ->  A. f 
g  e.  C )
32nfi 1606 . . . 4  |-  F/ f  g  e.  C
4 nfv 1683 . . . 4  |-  F/ f Fun  g
53, 4nfim 1867 . . 3  |-  F/ f ( g  e.  C  ->  Fun  g )
6 eleq1 2539 . . . 4  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
7 funeq 5607 . . . 4  |-  ( f  =  g  ->  ( Fun  f  <->  Fun  g ) )
86, 7imbi12d 320 . . 3  |-  ( f  =  g  ->  (
( f  e.  C  ->  Fun  f )  <->  ( g  e.  C  ->  Fun  g
) ) )
91bnj1436 32995 . . . . . 6  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
109bnj1299 32974 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
11 fnfun 5678 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1210, 11bnj31 32870 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1312bnj1265 32968 . . 3  |-  ( f  e.  C  ->  Fun  f )
145, 8, 13chvar 1982 . 2  |-  ( g  e.  C  ->  Fun  g )
1514rgen 2824 1  |-  A. g  e.  C  Fun  g
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815    C_ wss 3476   <.cop 4033    |` cres 5001   Fun wfun 5582    Fn wfn 5583   ` cfv 5588    predc-bnj14 32838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-in 3483  df-ss 3490  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-fun 5590  df-fn 5591
This theorem is referenced by:  bnj60  33215
  Copyright terms: Public domain W3C validator