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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 7206 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | 1 | rneqi 5801 | . 2 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | rnopab 5820 | . 2 ⊢ ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2168 | . . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 5348 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3508 | . . . . . 6 ⊢ ∃𝑤 𝑤 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1946 | . . . . . 6 ⊢ (∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑤 𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
8 | 6, 7 | mpbiran 707 | . . . . 5 ⊢ (∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜑) |
9 | 8 | 2exbii 1845 | . . . 4 ⊢ (∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
10 | 4, 9 | bitri 277 | . . 3 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦𝜑) |
11 | 10 | abbii 2886 | . 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
12 | 2, 3, 11 | 3eqtri 2848 | 1 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 {cab 2799 〈cop 4566 {copab 5120 ran crn 5550 {coprab 7151 |
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 df-oprab 7154 |
This theorem is referenced by: rnoprab2 7252 elrnmpores 7282 ellines 33608 |
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