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| Mirrors > Home > MPE Home > Th. List > rnoprab | Structured version Visualization version GIF version | ||
| Description: The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.) |
| Ref | Expression |
|---|---|
| rnoprab | ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfoprab2 7491 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 2 | 1 | rneqi 5948 | . 2 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
| 3 | rnopab 5965 | . 2 ⊢ ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 4 | exrot3 2165 | . . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
| 5 | opex 5469 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 6 | 5 | isseti 3498 | . . . . . 6 ⊢ ∃𝑤 𝑤 = 〈𝑥, 𝑦〉 |
| 7 | 19.41v 1949 | . . . . . 6 ⊢ (∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑤 𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
| 8 | 6, 7 | mpbiran 709 | . . . . 5 ⊢ (∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
| 9 | 8 | 2exbii 1849 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
| 10 | 4, 9 | bitri 275 | . . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
| 11 | 10 | abbii 2809 | . 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
| 12 | 2, 3, 11 | 3eqtri 2769 | 1 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 {cab 2714 〈cop 4632 {copab 5205 ran crn 5686 {coprab 7432 |
| 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 df-oprab 7435 |
| This theorem is referenced by: rnoprab2 7539 elrnmpores 7571 ellines 36153 |
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