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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 2349 | . . . 4 ⊢ Ⅎy∀y ∈ B (φ → x = 𝐶) | |
2 | 1 | nfmo 1917 | . . 3 ⊢ Ⅎy∃*x∀y ∈ B (φ → x = 𝐶) |
3 | rsp 2363 | . . . . . . . 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 1751 | . . . . 5 ⊢ ((y ∈ B ∧ φ) → ∀x(∀y ∈ B (φ → x = 𝐶) → x = 𝐶)) |
7 | moeq 2710 | . . . . 5 ⊢ ∃*x x = 𝐶 | |
8 | moim 1961 | . . . . 5 ⊢ (∀x(∀y ∈ B (φ → x = 𝐶) → x = 𝐶) → (∃*x x = 𝐶 → ∃*x∀y ∈ B (φ → x = 𝐶))) | |
9 | 6, 7, 8 | mpisyl 1332 | . . . 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 2420 | . 2 ⊢ (∃y ∈ B φ → ∃*x∀y ∈ B (φ → x = 𝐶)) |
12 | mormo 2515 | . 2 ⊢ (∃*x∀y ∈ B (φ → x = 𝐶) → ∃*x ∈ A ∀y ∈ B (φ → x = 𝐶)) | |
13 | reu5 2516 | . . 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 829 | . 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 1240 = wceq 1242 ∈ wcel 1390 ∃*wmo 1898 ∀wral 2300 ∃wrex 2301 ∃!wreu 2302 ∃*wrmo 2303 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-ral 2305 df-rex 2306 df-reu 2307 df-rmo 2308 df-v 2553 |
This theorem is referenced by: (None) |
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