Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabdm | Structured version Visualization version GIF version |
Description: Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
opabdm | ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 5535 | . 2 ⊢ dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} | |
2 | nfopab1 5099 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfeq2 2916 | . . 3 ⊢ Ⅎ𝑥 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | nfopab2 5100 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 4 | nfeq2 2916 | . . . 4 ⊢ Ⅎ𝑦 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | df-br 5031 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | eleq2 2821 | . . . . . 6 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
8 | opabidw 5380 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | bitrdi 290 | . . . . 5 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 𝜑)) |
10 | 6, 9 | syl5bb 286 | . . . 4 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑥𝑅𝑦 ↔ 𝜑)) |
11 | 5, 10 | exbid 2225 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (∃𝑦 𝑥𝑅𝑦 ↔ ∃𝑦𝜑)) |
12 | 3, 11 | abbid 2804 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} = {𝑥 ∣ ∃𝑦𝜑}) |
13 | 1, 12 | syl5eq 2785 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃wex 1786 ∈ wcel 2114 {cab 2716 〈cop 4522 class class class wbr 5030 {copab 5092 dom cdm 5525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-dm 5535 |
This theorem is referenced by: fpwrelmapffslem 30642 |
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