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Mirrors > Home > MPE Home > Th. List > ex-opab | Structured version Visualization version GIF version |
Description: Example for df-opab 4713. 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 11095 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 4cn 11098 | . . 3 ⊢ 4 ∈ ℂ | |
3 | 3p1e4 11153 | . . 3 ⊢ (3 + 1) = 4 | |
4 | 1 | elexi 3213 | . . . 4 ⊢ 3 ∈ V |
5 | 2 | elexi 3213 | . . . 4 ⊢ 4 ∈ V |
6 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 3 → (𝑥 ∈ ℂ ↔ 3 ∈ ℂ)) | |
7 | oveq1 6657 | . . . . . 6 ⊢ (𝑥 = 3 → (𝑥 + 1) = (3 + 1)) | |
8 | 7 | eqeq1d 2624 | . . . . 5 ⊢ (𝑥 = 3 → ((𝑥 + 1) = 𝑦 ↔ (3 + 1) = 𝑦)) |
9 | 6, 8 | 3anbi13d 1401 | . . . 4 ⊢ (𝑥 = 3 → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦))) |
10 | eleq1 2689 | . . . . 5 ⊢ (𝑦 = 4 → (𝑦 ∈ ℂ ↔ 4 ∈ ℂ)) | |
11 | eqeq2 2633 | . . . . 5 ⊢ (𝑦 = 4 → ((3 + 1) = 𝑦 ↔ (3 + 1) = 4)) | |
12 | 10, 11 | 3anbi23d 1402 | . . . 4 ⊢ (𝑦 = 4 → ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (3 + 1) = 𝑦) ↔ (3 ∈ ℂ ∧ 4 ∈ ℂ ∧ (3 + 1) = 4))) |
13 | eqid 2622 | . . . 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 4998 | . . 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 1244 | . 2 ⊢ 3{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)}4 |
16 | breq 4655 | . 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 248 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 {copab 4712 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 3c3 11071 4c4 11072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 df-ov 6653 df-2 11079 df-3 11080 df-4 11081 |
This theorem is referenced by: (None) |
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