| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ex-opab | Structured version Visualization version GIF version | ||
| Description: Example for df-opab 5206. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.) |
| Ref | Expression |
|---|---|
| ex-opab | ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3cn 12347 | . . 3 ⊢ 3 ∈ ℂ | |
| 2 | 4cn 12351 | . . 3 ⊢ 4 ∈ ℂ | |
| 3 | 3p1e4 12411 | . . 3 ⊢ (3 + 1) = 4 | |
| 4 | 1 | elexi 3503 | . . . 4 ⊢ 3 ∈ V |
| 5 | 2 | elexi 3503 | . . . 4 ⊢ 4 ∈ V |
| 6 | eleq1 2829 | . . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ)) | |
| 7 | oveq1 7438 | . . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1)) | |
| 8 | 7 | eqeq1d 2739 | . . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦)) |
| 9 | 6, 8 | 3anbi13d 1440 | . . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦))) |
| 10 | eleq1 2829 | . . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ)) | |
| 11 | eqeq2 2749 | . . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4)) | |
| 12 | 10, 11 | 3anbi23d 1441 | . . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))) |
| 13 | eqid 2737 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} | |
| 14 | 4, 5, 9, 12, 13 | brab 5548 | . . 3 ⊢ (3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4)) |
| 15 | 1, 2, 3, 14 | mpbir3an 1342 | . 2 ⊢ 3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 |
| 16 | breq 5145 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → (3𝑅4 ↔ 3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4)) | |
| 17 | 15, 16 | mpbiri 258 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 {copab 5205 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 3c3 12322 4c4 12323 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 df-ov 7434 df-2 12329 df-3 12330 df-4 12331 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |