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Theorem isarep2 5533
 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 5531. (Contributed by NM, 26-Oct-2006.)
Hypotheses
Ref Expression
isarep2.1
isarep2.2
Assertion
Ref Expression
isarep2
Distinct variable groups:   ,,,   ,   ,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem isarep2
StepHypRef Expression
1 resima 5178 . . . 4
2 resopab 5187 . . . . 5
32imaeq1i 5200 . . . 4
41, 3eqtr3i 2458 . . 3
5 funopab 5486 . . . . 5
6 isarep2.2 . . . . . . . 8
76rspec 2770 . . . . . . 7
8 nfv 1629 . . . . . . . 8
98mo3 2312 . . . . . . 7
107, 9sylibr 204 . . . . . 6
11 moanimv 2339 . . . . . 6
1210, 11mpbir 201 . . . . 5
135, 12mpgbir 1559 . . . 4
14 isarep2.1 . . . . 5
1514funimaex 5531 . . . 4
1613, 15ax-mp 8 . . 3
174, 16eqeltri 2506 . 2
1817isseti 2962 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wceq 1652  wsb 1658   wcel 1725  wmo 2282  wral 2705  cvv 2956  copab 4265   cres 4880  cima 4881   wfun 5448 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456
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