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Theorem isarep2 5533
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 5531. (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 5178 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ph } " A )
2 resopab 5187 . . . . 5  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
32imaeq1i 5200 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)
41, 3eqtr3i 2458 . . 3  |-  ( {
<. x ,  y >.  |  ph } " A
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )
5 funopab 5486 . . . . 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 2770 . . . . . . 7  |-  ( x  e.  A  ->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
8 nfv 1629 . . . . . . . 8  |-  F/ z
ph
98mo3 2312 . . . . . . 7  |-  ( E* y ph  <->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
107, 9sylibr 204 . . . . . 6  |-  ( x  e.  A  ->  E* y ph )
11 moanimv 2339 . . . . . 6  |-  ( E* y ( x  e.  A  /\  ph )  <->  ( x  e.  A  ->  E* y ph ) )
1210, 11mpbir 201 . . . . 5  |-  E* y
( x  e.  A  /\  ph )
135, 12mpgbir 1559 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
14 isarep2.1 . . . . 5  |-  A  e. 
_V
1514funimaex 5531 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)  e.  _V )
1613, 15ax-mp 8 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )  e. 
_V
174, 16eqeltri 2506 . 2  |-  ( {
<. x ,  y >.  |  ph } " A
)  e.  _V
1817isseti 2962 1  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   [wsb 1658    e. wcel 1725   E*wmo 2282   A.wral 2705   _Vcvv 2956   {copab 4265    |` cres 4880   "cima 4881   Fun wfun 5448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456
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