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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
StepHypRef Expression
1 resima 5311 . . . 4
2 resopab 5325 . . . . 5
32imaeq1i 5339 . . . 4
41, 3eqtr3i 2488 . . 3
5 funopab 5626 . . . . 5
6 isarep2.2 . . . . . . . 8
76rspec 2825 . . . . . . 7
8 nfv 1707 . . . . . . . 8
98mo3 2323 . . . . . . 7
107, 9sylibr 212 . . . . . 6
11 moanimv 2352 . . . . . 6
1210, 11mpbir 209 . . . . 5
135, 12mpgbir 1622 . . . 4
14 isarep2.1 . . . . 5
1514funimaex 5671 . . . 4
1613, 15ax-mp 5 . . 3
174, 16eqeltri 2541 . 2
1817isseti 3115 1
 Colors of variables: wff setvar class 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  Fun`wfun 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
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