Theorem isarep2 5673

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 5671. (Contributed by NM, 26-Oct-2006.)

Hypotheses

Ref	Expression
isarep2.1
isarep2.2

Assertion

Ref	Expression
isarep2

Distinct variable groups:   ,,,   ,   ,   ,

Proof of Theorem isarep2

Step	Hyp	Ref	Expression
1		resima 5311	. . . 4
2		resopab 5325	. . . . 5
3	2	imaeq1i 5339	. . . 4
4	1, 3	eqtr3i 2488	. . 3
5		funopab 5626	. . . . 5
6		isarep2.2	. . . . . . . 8
7	6	rspec 2825	. . . . . . 7
8		nfv 1707	. . . . . . . 8
9	8	mo3 2323	. . . . . . 7
10	7, 9	sylibr 212	. . . . . 6
11		moanimv 2352	. . . . . 6
12	10, 11	mpbir 209	. . . . 5
13	5, 12	mpgbir 1622	. . . 4
14		isarep2.1	. . . . 5
15	14	funimaex 5671	. . . 4
16	13, 15	ax-mp 5	. . 3
17	4, 16	eqeltri 2541	. 2
18	17	isseti 3115	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  [wsb 1739  e.wcel 1818  E*wmo 2283  A.wral 2807  cvv 3109  {copab 4509  |`cres 5006  "cima 5007  Funwfun 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-fun 5595