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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 5902 | . 2 ⊢ ran 𝑅 = {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} | |
2 | nfopab2 5219 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfeq2 2921 | . . 3 ⊢ Ⅎ𝑦 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | nfopab1 5218 | . . . . 5 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 4 | nfeq2 2921 | . . . 4 ⊢ Ⅎ𝑥 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | df-br 5149 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
7 | eleq2 2828 | . . . . . 6 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
8 | opabidw 5534 | . . . . . 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 2221 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (∃𝑥 𝑥𝑅𝑦 ↔ ∃𝑥𝜑)) |
12 | 3, 11 | abbid 2808 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} = {𝑦 ∣ ∃𝑥𝜑}) |
13 | 1, 12 | eqtrid 2787 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 〈cop 4637 class class class wbr 5148 {copab 5210 ran crn 5690 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: fpwrelmapffslem 32750 |
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