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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 7226 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | rneqi 5780 | . 2 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | rnopab 5797 | . 2 ⊢ ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2173 | . . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 5322 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3413 | . . . . . 6 ⊢ ∃𝑤 𝑤 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1957 | . . . . . 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 1855 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
10 | 4, 9 | bitri 278 | . . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 10 | abbii 2803 | . 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
12 | 2, 3, 11 | 3eqtri 2765 | 1 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∃wex 1786 {cab 2716 〈cop 4522 {copab 5092 ran crn 5526 {coprab 7171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-cnv 5533 df-dm 5535 df-rn 5536 df-oprab 7174 |
This theorem is referenced by: rnoprab2 7272 elrnmpores 7303 ellines 34092 |
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