Theorem isarep2 5346

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 5344. (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

Step	Hyp	Ref	Expression
1		resima 4980	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ph } " A )
2		resopab 4991	. . . . 5 |- ( { <. x , y >. | ph } |` A ) = { <. x , y >. | ( x e. A /\ ph ) }
3	2	imaeq1i 5007	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
4	1, 3	eqtr3i 2219	. . 3 |- ( { <. x , y >. | ph } " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
5		funopab 5294	. . . . 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 )
7	6	rspec 2549	. . . . . . 7 |- ( x e. A -> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
8		nfv 1542	. . . . . . . 8 |- F/ z ph
9	8	mo3 2099	. . . . . . 7 |- ( E* y ph <-> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
10	7, 9	sylibr 134	. . . . . 6 |- ( x e. A -> E* y ph )
11		moanimv 2120	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
12	10, 11	mpbir 146	. . . . 5 |- E* y ( x e. A /\ ph )
13	5, 12	mpgbir 1467	. . . 4 |- Fun { <. x , y >. | ( x e. A /\ ph ) }
14		isarep2.1	. . . . 5 |- A e. _V
15	14	funimaex 5344	. . . 4 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } -> ( { <. x , y >. | ( x e. A /\ ph ) } " A ) e. _V )
16	13, 15	ax-mp 5	. . 3 |- ( { <. x , y >. | ( x e. A /\ ph ) } " A ) e. _V
17	4, 16	eqeltri 2269	. 2 |- ( { <. x , y >. | ph } " A ) e. _V
18	17	isseti 2771	1 |- E. w w = ( { <. x , y >. | ph } " A )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   E.wex 1506   [wsb 1776   E*wmo 2046   e. wcel 2167   A.wral 2475   _Vcvv 2763   {copab 4094   |` cres 4666   "cima 4667   Fun wfun 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-fun 5261

This theorem is referenced by: (None)