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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 5123 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 5124 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfrnf 5819 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 5054 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabidw 5406 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 278 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1855 | . . 3 ⊢ (∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑) |
8 | 7 | abbii 2808 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑} |
9 | 3, 8 | eqtri 2765 | 1 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∃wex 1787 ∈ wcel 2110 {cab 2714 〈cop 4547 class class class wbr 5053 {copab 5115 ran crn 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-cnv 5559 df-dm 5561 df-rn 5562 |
This theorem is referenced by: rnmpt 5824 mptpreima 6101 rnoprab 7314 pwfir 8854 marypha2lem4 9054 hartogslem1 9158 axdc2lem 10062 abrexdomjm 30571 abrexexd 30573 lsmsnorb 31293 satfrnmapom 33045 rnttrcl 33521 rnmptsn 35243 abrexdom 35625 rncnvepres 36176 imaopab 39920 |
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