Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opabrn | Structured version Visualization version GIF version |
Description: Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
opabrn | ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 5753 | . 2 ⊢ ran 𝑅 = {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} | |
2 | nfopab2 5128 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfeq2 2995 | . . 3 ⊢ Ⅎ𝑦 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | nfopab1 5127 | . . . . 5 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 4 | nfeq2 2995 | . . . 4 ⊢ Ⅎ𝑥 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | df-br 5059 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | eleq2 2901 | . . . . . 6 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
8 | opabidw 5404 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
9 | 7, 8 | syl6bb 289 | . . . . 5 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 𝜑)) |
10 | 6, 9 | syl5bb 285 | . . . 4 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑥𝑅𝑦 ↔ 𝜑)) |
11 | 5, 10 | exbid 2221 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (∃𝑥 𝑥𝑅𝑦 ↔ ∃𝑥𝜑)) |
12 | 3, 11 | abbid 2887 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} = {𝑦 ∣ ∃𝑥𝜑}) |
13 | 1, 12 | syl5eq 2868 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 〈cop 4566 class class class wbr 5058 {copab 5120 ran crn 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: fpwrelmapffslem 30462 |
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