Theorem isarep2 6659

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 6656. (Contributed by NM, 26-Oct-2006.)

Hypotheses

Ref	Expression
isarep2.1	⊢ 𝐴 ∈ V
isarep2.2	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((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

Step	Hyp	Ref	Expression
1		resima 6035	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2		resopab 6054	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	imaeq1i 6077	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
4	1, 3	eqtr3i 2765	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
5		funopab 6603	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
6		isarep2.2	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)
7	6	rspec 3248	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8		nfv 1912	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
9	8	mo3 2562	. . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
10	7, 9	sylibr 234	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)
11		moanimv 2617	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
12	10, 11	mpbir 231	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
13	5, 12	mpgbir 1796	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		isarep2.1	. . . . 5 ⊢ 𝐴 ∈ V
15	14	funimaex 6656	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) ∈ V)
16	13, 15	ax-mp 5	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴) ∈ V
17	4, 16	eqeltri 2835	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
18	17	isseti 3496	1 ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1776  [wsb 2062   ∈ wcel 2106  ∃*wmo 2536  ∀wral 3059  Vcvv 3478  {copab 5210   ↾ cres 5691   “ cima 5692  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565

This theorem is referenced by: (None)