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Theorem isarep2 6431
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 6429. (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 5874 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2 resopab 5889 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32imaeq1i 5913 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
41, 3eqtr3i 2849 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
5 funopab 6378 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
6 isarep2.2 . . . . . . . 8 𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)
76rspec 3202 . . . . . . 7 (𝑥𝐴 → ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8 nfv 1916 . . . . . . . 8 𝑧𝜑
98mo3 2649 . . . . . . 7 (∃*𝑦𝜑 ↔ ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
107, 9sylibr 237 . . . . . 6 (𝑥𝐴 → ∃*𝑦𝜑)
11 moanimv 2707 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
1210, 11mpbir 234 . . . . 5 ∃*𝑦(𝑥𝐴𝜑)
135, 12mpgbir 1801 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
14 isarep2.1 . . . . 5 𝐴 ∈ V
1514funimaex 6429 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V)
1613, 15ax-mp 5 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V
174, 16eqeltri 2912 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
1817isseti 3494 1 𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wex 1781  [wsb 2070  wcel 2115  ∃*wmo 2622  wral 3133  Vcvv 3480  {copab 5114  cres 5544  cima 5545  Fun wfun 6337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-fun 6345
This theorem is referenced by: (None)
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