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Mirrors > Home > MPE Home > Th. List > rnopab | Structured version Visualization version GIF version |
Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
rnopab | ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopab1 4651 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 4652 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfrnf 5285 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 4584 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabid 4907 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 263 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1764 | . . 3 ⊢ (∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑) |
8 | 7 | abbii 2726 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑} |
9 | 3, 8 | eqtri 2632 | 1 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 〈cop 4131 class class class wbr 4583 {copab 4642 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: rnmpt 5292 mptpreima 5545 rnoprab 6641 marypha2lem4 8227 hartogslem1 8330 axdc2lem 9153 abrexdomjm 28729 abrexexd 28731 rnmptsn 32358 abrexdom 32695 |
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