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Theorem repizf2lem 4145
Description: Lemma for repizf2 4146. 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 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)

Proof of Theorem repizf2lem
StepHypRef Expression
1 df-mo 2023 . . . 4 (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑))
21imbi2i 225 . . 3 ((𝑥𝑤 → ∃*𝑦𝜑) ↔ (𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
32albii 1463 . 2 (∀𝑥(𝑥𝑤 → ∃*𝑦𝜑) ↔ ∀𝑥(𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
4 df-ral 2453 . 2 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥𝑤 → ∃*𝑦𝜑))
5 df-ral 2453 . . 3 (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑))
6 rabid 2645 . . . . . 6 (𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑} ↔ (𝑥𝑤 ∧ ∃𝑦𝜑))
76imbi1i 237 . . . . 5 ((𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ((𝑥𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑))
8 impexp 261 . . . . 5 (((𝑥𝑤 ∧ ∃𝑦𝜑) → ∃!𝑦𝜑) ↔ (𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
97, 8bitri 183 . . . 4 ((𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ (𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
109albii 1463 . . 3 (∀𝑥(𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑) ↔ ∀𝑥(𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
115, 10bitri 183 . 2 (∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑)))
123, 4, 113bitr4i 211 1 (∀𝑥𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346  wex 1485  ∃!weu 2019  ∃*wmo 2020  wcel 2141  wral 2448  {crab 2452
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-sb 1756  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-rab 2457
This theorem is referenced by:  repizf2  4146
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