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Theorem isarep2 6016
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 6014. (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 5466 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2 resopab 5481 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32imaeq1i 5498 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
41, 3eqtr3i 2675 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
5 funopab 5961 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
6 isarep2.2 . . . . . . . 8 𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)
76rspec 2960 . . . . . . 7 (𝑥𝐴 → ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8 nfv 1883 . . . . . . . 8 𝑧𝜑
98mo3 2536 . . . . . . 7 (∃*𝑦𝜑 ↔ ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
107, 9sylibr 224 . . . . . 6 (𝑥𝐴 → ∃*𝑦𝜑)
11 moanimv 2560 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
1210, 11mpbir 221 . . . . 5 ∃*𝑦(𝑥𝐴𝜑)
135, 12mpgbir 1766 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
14 isarep2.1 . . . . 5 𝐴 ∈ V
1514funimaex 6014 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V)
1613, 15ax-mp 5 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V
174, 16eqeltri 2726 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
1817isseti 3240 1 𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  [wsb 1937  wcel 2030  ∃*wmo 2499  wral 2941  Vcvv 3231  {copab 4745  cres 5145  cima 5146  Fun wfun 5920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928
This theorem is referenced by: (None)
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