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Mirrors > Home > MPE Home > Th. List > ex-opab | Structured version Visualization version GIF version |
Description: Example for df-opab 5129. 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 11719 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 4cn 11723 | . . 3 ⊢ 4 ∈ ℂ | |
3 | 3p1e4 11783 | . . 3 ⊢ (3 + 1) = 4 | |
4 | 1 | elexi 3513 | . . . 4 ⊢ 3 ∈ V |
5 | 2 | elexi 3513 | . . . 4 ⊢ 4 ∈ V |
6 | eleq1 2900 | . . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ)) | |
7 | oveq1 7163 | . . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1)) | |
8 | 7 | eqeq1d 2823 | . . . . 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 2900 | . . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ)) | |
11 | eqeq2 2833 | . . . . 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 2821 | . . . 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 5430 | . . 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 1337 | . 2 ⊢ 3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 |
16 | breq 5068 | . 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 260 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 {copab 5128 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 3c3 11694 4c4 11695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-iota 6314 df-fv 6363 df-ov 7159 df-2 11701 df-3 11702 df-4 11703 |
This theorem is referenced by: (None) |
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