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Theorem isarep2 5576
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 5574. (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 5218 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ph } " A )
2 resopab 5232 . . . . 5  |-  ( {
<. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
32imaeq1i 5246 . . . 4  |-  ( ( { <. x ,  y
>.  |  ph }  |`  A )
" A )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A
)
41, 3eqtr3i 2413 . . 3  |-  ( {
<. x ,  y >.  |  ph } " A
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  ph ) } " A )
5 funopab 5529 . . . . 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 2750 . . . . . . 7  |-  ( x  e.  A  ->  A. y A. z ( ( ph  /\ 
[ z  /  y ] ph )  ->  y  =  z ) )
8 nfv 1715 . . . . . . . 8  |-  F/ z
ph
98mo3 2258 . . . . . . 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 2283 . . . . . 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 1630 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
14 isarep2.1 . . . . 5  |-  A  e. 
_V
1514funimaex 5574 . . . 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 2466 . 2  |-  ( {
<. x ,  y >.  |  ph } " A
)  e.  _V
1817isseti 3040 1  |-  E. w  w  =  ( { <. x ,  y >.  |  ph } " A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397    = wceq 1399   E.wex 1620   [wsb 1747    e. wcel 1826   E*wmo 2219   A.wral 2732   _Vcvv 3034   {copab 4424    |` cres 4915   "cima 4916   Fun wfun 5490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-fun 5498
This theorem is referenced by: (None)
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