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Theorem isarep2 5666
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 5664. (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 5304 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ph } " A )
2 resopab 5318 . . . . 5  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
32imaeq1i 5332 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)
41, 3eqtr3i 2498 . . 3  |-  ( {
<. x ,  y >.  |  ph } " A
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )
5 funopab 5619 . . . . 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 2832 . . . . . . 7  |-  ( x  e.  A  ->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
8 nfv 1683 . . . . . . . 8  |-  F/ z
ph
98mo3 2320 . . . . . . 7  |-  ( E* y ph  <->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
107, 9sylibr 212 . . . . . 6  |-  ( x  e.  A  ->  E* y ph )
11 moanimv 2356 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
1210, 11mpbir 209 . . . . 5  |-  E* y
( x  e.  A  /\  ph )
135, 12mpgbir 1605 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
14 isarep2.1 . . . . 5  |-  A  e. 
_V
1514funimaex 5664 . . . 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 2551 . 2  |-  ( {
<. x ,  y >.  |  ph } " A
)  e.  _V
1817isseti 3119 1  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596   [wsb 1711    e. wcel 1767   E*wmo 2276   A.wral 2814   _Vcvv 3113   {copab 4504    |` cres 5001   "cima 5002   Fun wfun 5580
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588
This theorem is referenced by: (None)
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