Theorem isarep2 6523

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 6521. (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 5925	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2		resopab 5942	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	imaeq1i 5966	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
4	1, 3	eqtr3i 2768	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
5		funopab 6469	. . . . 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 3133	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
9	8	mo3 2564	. . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
10	7, 9	sylibr 233	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)
11		moanimv 2621	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
12	10, 11	mpbir 230	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
13	5, 12	mpgbir 1802	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		isarep2.1	. . . . 5 ⊢ 𝐴 ∈ V
15	14	funimaex 6521	. . . 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 3447	1 ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782  [wsb 2067   ∈ wcel 2106  ∃*wmo 2538  ∀wral 3064  Vcvv 3432  {copab 5136   ↾ cres 5591   “ cima 5592  Fun wfun 6427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435

This theorem is referenced by: (None)