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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 5218 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 5219 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfrnf 5964 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 5149 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabidw 5534 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 275 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1845 | . . 3 ⊢ (∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ ∃𝑥𝜑) |
8 | 7 | abbii 2807 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} = {𝑦 ∣ ∃𝑥𝜑} |
9 | 3, 8 | eqtri 2763 | 1 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 〈cop 4637 class class class wbr 5148 {copab 5210 ran crn 5690 |
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 |
This theorem is referenced by: rnopabss 5969 rnopab3 5970 rnmpt 5971 mptpreima 6260 rnoprab 7537 pwfir 9353 marypha2lem4 9476 hartogslem1 9580 rnttrcl 9760 axdc2lem 10486 abrexdomjm 32535 abrexexd 32537 lsmsnorb 33399 satfrnmapom 35355 rnmptsn 37318 abrexdom 37717 rncnvepres 38285 imaopab 42249 tfsconcatrn 43332 modelaxreplem3 44945 |
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