Theorem isarep2 5259

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 5257. (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 4901	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2		resopab 4912	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	imaeq1i 4927	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
4	1, 3	eqtr3i 2180	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
5		funopab 5207	. . . . 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 2509	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8		nfv 1508	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
9	8	mo3 2060	. . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
10	7, 9	sylibr 133	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)
11		moanimv 2081	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
12	10, 11	mpbir 145	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
13	5, 12	mpgbir 1433	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		isarep2.1	. . . . 5 ⊢ 𝐴 ∈ V
15	14	funimaex 5257	. . . 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 2230	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
18	17	isseti 2720	1 ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1333   = wceq 1335  ∃wex 1472  [wsb 1742  ∃*wmo 2007   ∈ wcel 2128  ∀wral 2435  Vcvv 2712  {copab 4026   ↾ cres 4590   “ cima 4591  Fun wfun 5166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 5174

This theorem is referenced by: (None)