Theorem isarep2 3564

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 3562.

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 3375	. . . 4 |- (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | ph}"A)
2		resopab 3379	. . . . 5 |- ({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)}
3		imaeq1 3385	. . . . 5 |- (({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)} -> (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | (x e. A /\ ph)}"A))
4	2, 3	ax-mp 7	. . . 4 |- (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | (x e. A /\ ph)}"A)
5	1, 4	eqtr3 1489	. . 3 |- ({<.x, y>. | ph}"A) = ({<.x, y>. | (x e. A /\ ph)}"A)
6		funopab 3534	. . . . 5 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
7		isarep2.2	. . . . . . . 8 |- A.x e. A A.yA.z((ph /\ [z / y]ph) -> y = z)
8	7	rspec 1689	. . . . . . 7 |- (x e. A -> A.yA.z((ph /\ [z / y]ph) -> y = z))
9		ax-17 968	. . . . . . . 8 |- (ph -> A.zph)
10	9	mo3 1394	. . . . . . 7 |- (E*yph <-> A.yA.z((ph /\ [z / y]ph) -> y = z))
11	8, 10	sylibr 200	. . . . . 6 |- (x e. A -> E*yph)
12		moanimv 1422	. . . . . 6 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
13	11, 12	mpbir 190	. . . . 5 |- E*y(x e. A /\ ph)
14	6, 13	mpgbir 985	. . . 4 |- Fun {<.x, y>. | (x e. A /\ ph)}
15		isarep2.1	. . . . 5 |- A e. V
16	15	funimaex 3562	. . . 4 |- (Fun {<.x, y>. | (x e. A /\ ph)} -> ({<.x, y>. | (x e. A /\ ph)}"A) e. V)
17	14, 16	ax-mp 7	. . 3 |- ({<.x, y>. | (x e. A /\ ph)}"A) e. V
18	5, 17	eqeltr 1536	. 2 |- ({<.x, y>. | ph}"A) e. V
19		isset 1805	. 2 |- (({<.x, y>. | ph}"A) e. V <-> E.w w = ({<.x, y>. | ph}"A))
20	18, 19	mpbi 189	1 |- E.w w = ({<.x, y>. | ph}"A)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  [wsbc 1166  E*wmo 1374  A.wral 1637  Vcvv 1802  {copab 2656   |` cres 3162  "cima 3163  Fun wfun 3166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182

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