Theorem isarep2 3584

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 3582.

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 3397	. . . 4 |- (({<.x, y>. | ph} |` A)"A) = ({<.x, y>. | ph}"A)
2		resopab 3401	. . . . 5 |- ({<.x, y>. | ph} |` A) = {<.x, y>. | (x e. A /\ ph)}
3		imaeq1 3407	. . . . 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 1500	. . 3 |- ({<.x, y>. | ph}"A) = ({<.x, y>. | (x e. A /\ ph)}"A)
6		funopab 3554	. . . . 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 1700	. . . . . . 7 |- (x e. A -> A.yA.z((ph /\ [z / y]ph) -> y = z))
9		ax-17 973	. . . . . . . 8 |- (ph -> A.zph)
10	9	mo3 1403	. . . . . . 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 1431	. . . . . 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 990	. . . 4 |- Fun {<.x, y>. | (x e. A /\ ph)}
15		isarep2.1	. . . . 5 |- A e. V
16	15	funimaex 3582	. . . 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 1547	. 2 |- ({<.x, y>. | ph}"A) e. V
19		isset 1817	. 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 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  E*wmo 1383  A.wral 1648  Vcvv 1814  {copab 2671   |` cres 3178  "cima 3179  Fun wfun 3182

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

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