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Theorem isarep2 4929
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 4927. (Contributed by NM, 26-Oct-2006.)
Hypotheses
Ref Expression
isarep2.1 A V
isarep2.2 x A yz((φ [z / y]φ) → y = z)
Assertion
Ref Expression
isarep2 w w = ({⟨x, y⟩ ∣ φ} “ A)
Distinct variable groups:   x,w,y,A   y,z   φ,w   φ,z
Allowed substitution hints:   φ(x,y)   A(z)

Proof of Theorem isarep2
StepHypRef Expression
1 resima 4586 . . . 4 (({⟨x, y⟩ ∣ φ} ↾ A) “ A) = ({⟨x, y⟩ ∣ φ} “ A)
2 resopab 4595 . . . . 5 ({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}
32imaeq1i 4608 . . . 4 (({⟨x, y⟩ ∣ φ} ↾ A) “ A) = ({⟨x, y⟩ ∣ (x A φ)} “ A)
41, 3eqtr3i 2059 . . 3 ({⟨x, y⟩ ∣ φ} “ A) = ({⟨x, y⟩ ∣ (x A φ)} “ A)
5 funopab 4878 . . . . 5 (Fun {⟨x, y⟩ ∣ (x A φ)} ↔ x∃*y(x A φ))
6 isarep2.2 . . . . . . . 8 x A yz((φ [z / y]φ) → y = z)
76rspec 2367 . . . . . . 7 (x Ayz((φ [z / y]φ) → y = z))
8 nfv 1418 . . . . . . . 8 zφ
98mo3 1951 . . . . . . 7 (∃*yφyz((φ [z / y]φ) → y = z))
107, 9sylibr 137 . . . . . 6 (x A∃*yφ)
11 moanimv 1972 . . . . . 6 (∃*y(x A φ) ↔ (x A∃*yφ))
1210, 11mpbir 134 . . . . 5 ∃*y(x A φ)
135, 12mpgbir 1339 . . . 4 Fun {⟨x, y⟩ ∣ (x A φ)}
14 isarep2.1 . . . . 5 A V
1514funimaex 4927 . . . 4 (Fun {⟨x, y⟩ ∣ (x A φ)} → ({⟨x, y⟩ ∣ (x A φ)} “ A) V)
1613, 15ax-mp 7 . . 3 ({⟨x, y⟩ ∣ (x A φ)} “ A) V
174, 16eqeltri 2107 . 2 ({⟨x, y⟩ ∣ φ} “ A) V
1817isseti 2557 1 w w = ({⟨x, y⟩ ∣ φ} “ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378   wcel 1390  [wsb 1642  ∃*wmo 1898  wral 2300  Vcvv 2551  {copab 3808  cres 4290  cima 4291  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by: (None)
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