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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 5951 | . 2 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
3 | rnopab 5968 | . 2 ⊢ ran {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
4 | exrot3 2163 | . . . 4 ⊢ (∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑤(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | opex 5475 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
6 | 5 | isseti 3496 | . . . . . 6 ⊢ ∃𝑤 𝑤 = 〈𝑥, 𝑦〉 |
7 | 19.41v 1947 | . . . . . 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 1846 | . . . 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 2807 | . 2 ⊢ {𝑧 ∣ ∃𝑤∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
12 | 2, 3, 11 | 3eqtri 2767 | 1 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 {cab 2712 〈cop 4637 {copab 5210 ran crn 5690 {coprab 7432 |
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 df-oprab 7435 |
This theorem is referenced by: rnoprab2 7538 elrnmpores 7571 ellines 36134 |
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