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 4644. 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 10972 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 4cn 10975 | . . 3 ⊢ 4 ∈ ℂ | |
3 | 3p1e4 11030 | . . 3 ⊢ (3 + 1) = 4 | |
4 | 1 | elexi 3186 | . . . 4 ⊢ 3 ∈ V |
5 | 2 | elexi 3186 | . . . 4 ⊢ 4 ∈ V |
6 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ)) | |
7 | oveq1 6556 | . . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1)) | |
8 | 7 | eqeq1d 2612 | . . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦)) |
9 | 6, 8 | 3anbi13d 1393 | . . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦))) |
10 | eleq1 2676 | . . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ)) | |
11 | eqeq2 2621 | . . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4)) | |
12 | 10, 11 | 3anbi23d 1394 | . . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))) |
13 | eqid 2610 | . . . 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 4923 | . . 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 1237 | . 2 ⊢ 3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 |
16 | breq 4585 | . 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 247 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 {copab 4642 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 3c3 10948 4c4 10949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 df-ov 6552 df-2 10956 df-3 10957 df-4 10958 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |