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Theorem isarep2 6225
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 6223. (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 5682 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2 resopab 5698 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
32imaeq1i 5719 . . . 4 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
41, 3eqtr3i 2804 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴)
5 funopab 6172 . . . . 5 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
6 isarep2.2 . . . . . . . 8 𝑥𝐴𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)
76rspec 3113 . . . . . . 7 (𝑥𝐴 → ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8 nfv 1957 . . . . . . . 8 𝑧𝜑
98mo3 2580 . . . . . . 7 (∃*𝑦𝜑 ↔ ∀𝑦𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
107, 9sylibr 226 . . . . . 6 (𝑥𝐴 → ∃*𝑦𝜑)
11 moanimv 2654 . . . . . 6 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
1210, 11mpbir 223 . . . . 5 ∃*𝑦(𝑥𝐴𝜑)
135, 12mpgbir 1843 . . . 4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
14 isarep2.1 . . . . 5 𝐴 ∈ V
1514funimaex 6223 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V)
1613, 15ax-mp 5 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} “ 𝐴) ∈ V
174, 16eqeltri 2855 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
1817isseti 3411 1 𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599   = wceq 1601  wex 1823  [wsb 2011  wcel 2107  ∃*wmo 2549  wral 3090  Vcvv 3398  {copab 4950  cres 5359  cima 5360  Fun wfun 6131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-fun 6139
This theorem is referenced by: (None)
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