Theorem isarep2 5305

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 5303. (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 4942	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ph } " A )
2		resopab 4953	. . . . 5 |- ( { <. x , y >. | ph } |` A ) = { <. x , y >. | ( x e. A /\ ph ) }
3	2	imaeq1i 4969	. . . 4 |- ( ( { <. x , y >. | ph } |` A ) " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
4	1, 3	eqtr3i 2200	. . 3 |- ( { <. x , y >. | ph } " A ) = ( { <. x , y >. | ( x e. A /\ ph ) } " A )
5		funopab 5253	. . . . 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 2529	. . . . . . 7 |- ( x e. A -> A. y A. z ( ( ph /\ [ z / y ] ph ) -> y = z ) )
8		nfv 1528	. . . . . . . 8 |- F/ z ph
9	8	mo3 2080	. . . . . . 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 2101	. . . . . 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 1453	. . . 4 |- Fun { <. x , y >. | ( x e. A /\ ph ) }
14		isarep2.1	. . . . 5 |- A e. _V
15	14	funimaex 5303	. . . 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 2250	. 2 |- ( { <. x , y >. | ph } " A ) e. _V
18	17	isseti 2747	1 |- E. w w = ( { <. x , y >. | ph } " A )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   = wceq 1353   E.wex 1492   [wsb 1762   E*wmo 2027   e. wcel 2148   A.wral 2455   _Vcvv 2739   {copab 4065   |` cres 4630   "cima 4631   Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220

This theorem is referenced by: (None)