ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isarep2 Unicode version

Theorem isarep2 5178
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " [ i, 
[ i, i  ] => o  ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5176. (Contributed by NM, 26-Oct-2006.)
Hypotheses
Ref Expression
isarep2.1  |-  A  e. 
_V
isarep2.2  |-  A. x  e.  A  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z )
Assertion
Ref Expression
isarep2  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Distinct variable groups:    x, w, y, A    y, z    ph, w    ph, z
Allowed substitution hints:    ph( x, y)    A( z)

Proof of Theorem isarep2
StepHypRef Expression
1 resima 4820 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ph } " A )
2 resopab 4831 . . . . 5  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
32imaeq1i 4846 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)
41, 3eqtr3i 2138 . . 3  |-  ( {
<. x ,  y >.  |  ph } " A
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )
5 funopab 5126 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) } 
<-> 
A. x E* y
( x  e.  A  /\  ph ) )
6 isarep2.2 . . . . . . . 8  |-  A. x  e.  A  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z )
76rspec 2459 . . . . . . 7  |-  ( x  e.  A  ->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
8 nfv 1491 . . . . . . . 8  |-  F/ z
ph
98mo3 2029 . . . . . . 7  |-  ( E* y ph  <->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
107, 9sylibr 133 . . . . . 6  |-  ( x  e.  A  ->  E* y ph )
11 moanimv 2050 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
1210, 11mpbir 145 . . . . 5  |-  E* y
( x  e.  A  /\  ph )
135, 12mpgbir 1412 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
14 isarep2.1 . . . . 5  |-  A  e. 
_V
1514funimaex 5176 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)  e.  _V )
1613, 15ax-mp 5 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )  e. 
_V
174, 16eqeltri 2188 . 2  |-  ( {
<. x ,  y >.  |  ph } " A
)  e.  _V
1817isseti 2666 1  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   [wsb 1718   E*wmo 1976   A.wral 2391   _Vcvv 2658   {copab 3956    |` cres 4509   "cima 4510   Fun wfun 5085
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator