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Theorem isarep2 5087
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 5085. (Contributed by NM, 26-Oct-2006.)
Hypotheses
Ref Expression
isarep2.1 𝐴 ∈ V
isarep2.2 𝑥𝐴𝑦𝑧((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 4732 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2 resopab 4743 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32imaeq1i 4758 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
41, 3eqtr3i 2110 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
5 funopab 5035 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
6 isarep2.2 . . . . . . . 8 𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)
76rspec 2427 . . . . . . 7 (𝑥𝐴 → ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8 nfv 1466 . . . . . . . 8 𝑧𝜑
98mo3 2002 . . . . . . 7 (∃*𝑦𝜑 ↔ ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
107, 9sylibr 132 . . . . . 6 (𝑥𝐴 → ∃*𝑦𝜑)
11 moanimv 2023 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
1210, 11mpbir 144 . . . . 5 ∃*𝑦(𝑥𝐴𝜑)
135, 12mpgbir 1387 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
14 isarep2.1 . . . . 5 𝐴 ∈ V
1514funimaex 5085 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V)
1613, 15ax-mp 7 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V
174, 16eqeltri 2160 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
1817isseti 2627 1 𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  wcel 1438  [wsb 1692  ∃*wmo 1949  wral 2359  Vcvv 2619  {copab 3890  cres 4430  cima 4431  Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fun 5004
This theorem is referenced by: (None)
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