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Theorem isarep2 4986
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 4984. (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 4643 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ph } " A )
2 resopab 4652 . . . . 5  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
32imaeq1i 4665 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)
41, 3eqtr3i 2062 . . 3  |-  ( {
<. x ,  y >.  |  ph } " A
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )
5 funopab 4935 . . . . 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 2373 . . . . . . 7  |-  ( x  e.  A  ->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
8 nfv 1421 . . . . . . . 8  |-  F/ z
ph
98mo3 1954 . . . . . . 7  |-  ( E* y ph  <->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
107, 9sylibr 137 . . . . . 6  |-  ( x  e.  A  ->  E* y ph )
11 moanimv 1975 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
1210, 11mpbir 134 . . . . 5  |-  E* y
( x  e.  A  /\  ph )
135, 12mpgbir 1342 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
14 isarep2.1 . . . . 5  |-  A  e. 
_V
1514funimaex 4984 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)  e.  _V )
1613, 15ax-mp 7 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )  e. 
_V
174, 16eqeltri 2110 . 2  |-  ( {
<. x ,  y >.  |  ph } " A
)  e.  _V
1817isseti 2563 1  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   [wsb 1645   E*wmo 1901   A.wral 2306   _Vcvv 2557   {copab 3817    |` cres 4347   "cima 4348   Fun wfun 4896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904
This theorem is referenced by: (None)
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