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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 5685 | . 2 ⊢ dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} | |
2 | nfopab1 5217 | . . . 4 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfeq2 2921 | . . 3 ⊢ Ⅎ𝑥 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | nfopab2 5218 | . . . . 5 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 4 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑦 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | df-br 5148 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | eleq2 2823 | . . . . . 6 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
8 | opabidw 5523 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | bitrdi 287 | . . . . 5 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 𝜑)) |
10 | 6, 9 | bitrid 283 | . . . 4 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑥𝑅𝑦 ↔ 𝜑)) |
11 | 5, 10 | exbid 2217 | . . 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 | eqtrid 2785 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 〈cop 4633 class class class wbr 5147 {copab 5209 dom cdm 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-dm 5685 |
This theorem is referenced by: fpwrelmapffslem 31935 |
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