Theorem isarep2 5006

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 5004. (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 4661	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ph } " A )
2		resopab 4672	. . . . 5 |- ( { <. x , y >. | ph } |` A ) = { <. x , y >. | ( x e. A /\ ph ) }
3	2	imaeq1i 4685	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
4	1, 3	eqtr3i 2103	. . 3 |- ( { <. x , y >. | ph } " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
5		funopab 4955	. . . . 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 2415	. . . . . . 7 |- ( x e. A -> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
8		nfv 1461	. . . . . . . 8 |- F/ z ph
9	8	mo3 1995	. . . . . . 7 |- ( E* y ph <-> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
10	7, 9	sylibr 132	. . . . . 6 |- ( x e. A -> E* y ph )
11		moanimv 2016	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
12	10, 11	mpbir 144	. . . . 5 |- E* y ( x e. A /\ ph )
13	5, 12	mpgbir 1382	. . . 4 |- Fun { <. x , y >. | ( x e. A /\ ph ) }
14		isarep2.1	. . . . 5 |- A e. _V
15	14	funimaex 5004	. . . 4 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } -> ( { <. x , y >. | ( x e. A /\ ph ) } " A ) e. _V )
16	13, 15	ax-mp 7	. . 3 |- ( { <. x , y >. | ( x e. A /\ ph ) } " A ) e. _V
17	4, 16	eqeltri 2151	. 2 |- ( { <. x , y >. | ph } " A ) e. _V
18	17	isseti 2607	1 |- E. w w = ( { <. x , y >. | ph } " A )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   [wsb 1685   E*wmo 1942   A.wral 2348   _Vcvv 2601   {copab 3838   |` cres 4365   "cima 4366   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924

This theorem is referenced by: (None)