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Mirrors > Home > MPE Home > Th. List > ex-opab | Structured version Visualization version GIF version |
Description: Example for df-opab 5215. 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 12331 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 4cn 12335 | . . 3 ⊢ 4 ∈ ℂ | |
3 | 3p1e4 12395 | . . 3 ⊢ (3 + 1) = 4 | |
4 | 1 | elexi 3493 | . . . 4 ⊢ 3 ∈ V |
5 | 2 | elexi 3493 | . . . 4 ⊢ 4 ∈ V |
6 | eleq1 2817 | . . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ)) | |
7 | oveq1 7433 | . . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1)) | |
8 | 7 | eqeq1d 2730 | . . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦)) |
9 | 6, 8 | 3anbi13d 1434 | . . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦))) |
10 | eleq1 2817 | . . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ)) | |
11 | eqeq2 2740 | . . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4)) | |
12 | 10, 11 | 3anbi23d 1435 | . . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))) |
13 | eqid 2728 | . . . 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 5549 | . . 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 1338 | . 2 ⊢ 3{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 |
16 | breq 5154 | . 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 257 | 1 ⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 {copab 5214 (class class class)co 7426 ℂcc 11144 1c1 11147 + caddc 11149 3c3 12306 4c4 12307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-1cn 11204 ax-addcl 11206 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-iota 6505 df-fv 6561 df-ov 7429 df-2 12313 df-3 12314 df-4 12315 |
This theorem is referenced by: (None) |
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