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Mirrors > Home > ILE Home > Th. List > repizf2lem | GIF version |
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.) |
Ref | Expression |
---|---|
repizf2lem | ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2023 | . . . 4 ⊢ (∃*𝑦𝜑 ↔ (∃𝑦𝜑 → ∃!𝑦𝜑)) | |
2 | 1 | imbi2i 225 | . . 3 ⊢ ((𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ (𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑))) |
3 | 2 | albii 1463 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑤 → (∃𝑦𝜑 → ∃!𝑦𝜑))) |
4 | df-ral 2453 | . 2 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑤 → ∃*𝑦𝜑)) | |
5 | df-ral 2453 | . . 3 ⊢ (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → ∃!𝑦𝜑)) | |
6 | rabid 2645 | . . . . . 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 1463 | . . 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) |
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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