Theorem repizf2lem 4093

Description: Lemma for repizf2 4094. If we have a function-like proposition which provides at most one value of 𝑦 for each 𝑥 in a set 𝑤, we can change "at most one" to "exactly one" by restricting the values of 𝑥 to those values for which the proposition provides a value of 𝑦. (Contributed by Jim Kingdon, 7-Sep-2018.)

Assertion

Ref	Expression
repizf2lem	⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Proof of Theorem repizf2lem

Step	Hyp	Ref	Expression
1		df-mo 2004	. . . 4 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2	1	imbi2i 225	. . 3 ⊢ ((𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
3	2	albii 1447	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
4		df-ral 2422	. 2 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑))
5		df-ral 2422	. . 3 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑))
6		rabid 2609	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ↔ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑))
7	6	imbi1i 237	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑))
8		impexp 261	. . . . 5 ⊢ (((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
9	7, 8	bitri 183	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
10	9	albii 1447	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
11	5, 10	bitri 183	. 2 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
12	3, 4, 11	3bitr4i 211	1 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃wex 1469   ∈ wcel 1481  ∃!weu 2000  ∃*wmo 2001  ∀wral 2417  {crab 2421

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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-rab 2426

This theorem is referenced by:  repizf2  4094