| 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 5899 | . 2 ⊢ ran 𝑅 = {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} | |
| 2 | nfopab2 5214 | . . . 4 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 3 | 2 | nfeq2 2923 | . . 3 ⊢ Ⅎ𝑦 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
| 4 | nfopab1 5213 | . . . . 5 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 5 | 4 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑥 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
| 6 | df-br 5144 | . . . . 5 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
| 7 | eleq2 2830 | . . . . . 6 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (〈𝑥, 𝑦〉 ∈ 𝑅 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
| 8 | opabidw 5529 | . . . . . 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 2223 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (∃𝑥 𝑥𝑅𝑦 ↔ ∃𝑥𝜑)) |
| 12 | 3, 11 | abbid 2810 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → {𝑦 ∣ ∃𝑥 𝑥𝑅𝑦} = {𝑦 ∣ ∃𝑥𝜑}) |
| 13 | 1, 12 | eqtrid 2789 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 〈cop 4632 class class class wbr 5143 {copab 5205 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: fpwrelmapffslem 32743 |
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