Theorem isarep2 4499

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 4496.

Hypotheses

Ref	Expression
isarep2.1	|- A e. _V
isarep2.2	|- A.x e. A A.yA.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

Proof of Theorem isarep2

Step	Hyp	Ref	Expression
1		resima 4247	. . . 4 |- (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | ph}"A)
2		resopab 4252	. . . . 5 |- ({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)}
3	2	imaeq1i 4261	. . . 4 |- (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | (x e. A /\ ph)}"A)
4	1, 3	eqtr3i 1910	. . 3 |- ({<.x, y>. | ph}"A) = ({<.x, y>. | (x e. A /\ ph)}"A)
5		funopab 4455	. . . . 5 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
6		isarep2.2	. . . . . . . 8 |- A.x e. A A.yA.z((ph /\ [z / y]ph) -> y = z)
7	6	rspec 2158	. . . . . . 7 |- (x e. A -> A.yA.z((ph /\ [z / y]ph) -> y = z))
8		ax-17 1317	. . . . . . . 8 |- (ph -> A.zph)
9	8	mo3 1797	. . . . . . 7 |- (E*yph <-> A.yA.z((ph /\ [z / y]ph) -> y = z))
10	7, 9	sylibr 217	. . . . . 6 |- (x e. A -> E*yph)
11		moanimv 1829	. . . . . 6 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
12	10, 11	mpbir 207	. . . . 5 |- E*y(x e. A /\ ph)
13	5, 12	mpgbir 1334	. . . 4 |- Fun {<.x, y>. | (x e. A /\ ph)}
14		isarep2.1	. . . . 5 |- A e. _V
15	14	funimaex 4496	. . . 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 1967	. 2 |- ({<.x, y>. | ph}"A) e. _V
18		isset 2296	. 2 |- (({<.x, y>. | ph}"A) e. _V <-> E.w w = ({<.x, y>. | ph}"A))
19	17, 18	mpbi 206	1 |- E.w w = ({<.x, y>. | ph}"A)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E*wmo 1772  A.wral 2105  _Vcvv 2292  {copab 3395   |` cres 3988  "cima 3989  Fun wfun 3992

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008

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