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Mirrors > Home > ILE Home > Th. List > reusv1 | GIF version |
Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
reusv1 | ⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 2355 | . . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) | |
2 | 1 | nfmo 1920 | . . 3 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) |
3 | rsp 2369 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶))) | |
4 | 3 | impd 242 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶)) |
5 | 4 | com12 27 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)) |
6 | 5 | alrimiv 1754 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)) |
7 | moeq 2716 | . . . . 5 ⊢ ∃*𝑥 𝑥 = 𝐶 | |
8 | moim 1964 | . . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶) → (∃*𝑥 𝑥 = 𝐶 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | |
9 | 6, 7, 8 | mpisyl 1335 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
10 | 9 | ex 108 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
11 | 2, 10 | rexlimi 2426 | . 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
12 | mormo 2521 | . 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) | |
13 | reu5 2522 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) | |
14 | 13 | rbaib 830 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
15 | 11, 12, 14 | 3syl 17 | 1 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∃*wmo 1901 ∀wral 2306 ∃wrex 2307 ∃!wreu 2308 ∃*wrmo 2309 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-v 2559 |
This theorem is referenced by: (None) |
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