Theorem isarep2 5443

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 5441. (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 5071	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ph } " A )
2		resopab 5082	. . . . 5 |- ( { <. x , y >. | ph } |` A ) = { <. x , y >. | ( x e. A /\ ph ) }
3	2	imaeq1i 5098	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
4	1, 3	eqtr3i 2255	. . 3 |- ( { <. x , y >. | ph } " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
5		funopab 5387	. . . . 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 2594	. . . . . . 7 |- ( x e. A -> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
8		nfv 1577	. . . . . . . 8 |- F/ z ph
9	8	mo3 2135	. . . . . . 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 2156	. . . . . 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 1502	. . . 4 |- Fun { <. x , y >. | ( x e. A /\ ph ) }
14		isarep2.1	. . . . 5 |- A e. _V
15	14	funimaex 5441	. . . 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 2305	. 2 |- ( { <. x , y >. | ph } " A ) e. _V
18	17	isseti 2822	1 |- E. w w = ( { <. x , y >. | ph } " A )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   E.wex 1541   [wsb 1811   E*wmo 2081   e. wcel 2203   A.wral 2520   _Vcvv 2813   {copab 4170   |` cres 4751   "cima 4752   Fun wfun 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354

This theorem is referenced by: (None)