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Theorem isarep2 4912
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 4910. (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 4570 . . . 4 (({⟨x, y⟩ ∣ φ} ↾ A) “ A) = ({⟨x, y⟩ ∣ φ} “ A)
2 resopab 4579 . . . . 5 ({⟨x, y⟩ ∣ φ} ↾ A) = {⟨x, y⟩ ∣ (x A φ)}
32imaeq1i 4592 . . . 4 (({⟨x, y⟩ ∣ φ} ↾ A) “ A) = ({⟨x, y⟩ ∣ (x A φ)} “ A)
41, 3eqtr3i 2044 . . 3 ({⟨x, y⟩ ∣ φ} “ A) = ({⟨x, y⟩ ∣ (x A φ)} “ A)
5 funopab 4861 . . . . 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 2351 . . . . . . 7 (x Ayz((φ [z / y]φ) → y = z))
8 nfv 1402 . . . . . . . 8 zφ
98mo3 1936 . . . . . . 7 (∃*yφyz((φ [z / y]φ) → y = z))
107, 9sylibr 137 . . . . . 6 (x A∃*yφ)
11 moanimv 1957 . . . . . 6 (∃*y(x A φ) ↔ (x A∃*yφ))
1210, 11mpbir 134 . . . . 5 ∃*y(x A φ)
135, 12mpgbir 1322 . . . 4 Fun {⟨x, y⟩ ∣ (x A φ)}
14 isarep2.1 . . . . 5 A V
1514funimaex 4910 . . . 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 2092 . 2 ({⟨x, y⟩ ∣ φ} “ A) V
1817isseti 2541 1 w w = ({⟨x, y⟩ ∣ φ} “ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228  wex 1362   wcel 1374  [wsb 1627  ∃*wmo 1883  wral 2284  Vcvv 2535  {copab 3791  cres 4274  cima 4275  Fun wfun 4823
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-fun 4831
This theorem is referenced by: (None)
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