Theorem repizf2lem 4221

Description: Lemma for repizf2 4222. 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 2059	. . . 4 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2	1	imbi2i 226	. . 3 ⊢ ((𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
3	2	albii 1494	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
4		df-ral 2491	. 2 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑))
5		df-ral 2491	. . 3 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑))
6		rabid 2684	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ↔ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑))
7	6	imbi1i 238	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑))
8		impexp 263	. . . . 5 ⊢ (((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
9	7, 8	bitri 184	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
10	9	albii 1494	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
11	5, 10	bitri 184	. 2 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
12	3, 4, 11	3bitr4i 212	1 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  ∃wex 1516  ∃!weu 2055  ∃*wmo 2056   ∈ wcel 2178  ∀wral 2486  {crab 2490

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-rab 2495

This theorem is referenced by:  repizf2  4222