Theorem repizf2lem 3914

Description: Lemma for repizf2 3915. 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 1904	. . . 4 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
2	1	imbi2i 215	. . 3 ⊢ ((𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
3	2	albii 1359	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
4		df-ral 2311	. 2 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑))
5		df-ral 2311	. . 3 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑))
6		rabid 2485	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ↔ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑))
7	6	imbi1i 227	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑))
8		impexp 250	. . . . 5 ⊢ (((𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
9	7, 8	bitri 173	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
10	9	albii 1359	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
11	5, 10	bitri 173	. 2 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
12	3, 4, 11	3bitr4i 201	1 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  ∃*wmo 1901  ∀wral 2306  {crab 2310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rab 2315

This theorem is referenced by:  repizf2  3915