Theorem isarep2 5463

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 5461. (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 5091	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
2		resopab 5102	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
3	2	imaeq1i 5118	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
4	1, 3	eqtr3i 2261	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} “ 𝐴)
5		funopab 5407	. . . . 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 2602	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
8		nfv 1581	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
9	8	mo3 2141	. . . . . . 7 ⊢ (∃*𝑦𝜑 ↔ ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧))
10	7, 9	sylibr 134	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)
11		moanimv 2162	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
12	10, 11	mpbir 146	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
13	5, 12	mpgbir 1506	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
14		isarep2.1	. . . . 5 ⊢ 𝐴 ∈ V
15	14	funimaex 5461	. . . 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 2311	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ∈ V
18	17	isseti 2830	1 ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃wex 1545  [wsb 1815  ∃*wmo 2087   ∈ wcel 2209  ∀wral 2528  Vcvv 2821  {copab 4186   ↾ cres 4771   “ cima 4772  Fun wfun 5366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374

This theorem is referenced by: (None)