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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 5174 | . . 3 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | nfopab2 5175 | . . 3 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 1, 2 | dfrnf 5902 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦} |
4 | df-br 5105 | . . . . 5 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
5 | opabidw 5479 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | |
6 | 4, 5 | bitri 275 | . . . 4 ⊢ (𝑥{〈𝑥, 𝑦〉 ∣ 𝜑}𝑦 ↔ 𝜑) |
7 | 6 | exbii 1851 | . . 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 2766 | 1 ⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2715 〈cop 4591 class class class wbr 5104 {copab 5166 ran crn 5632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-cnv 5639 df-dm 5641 df-rn 5642 |
This theorem is referenced by: rnmpt 5907 mptpreima 6187 rnoprab 7453 pwfir 9054 marypha2lem4 9308 hartogslem1 9412 rnttrcl 9592 axdc2lem 10318 abrexdomjm 31237 abrexexd 31239 lsmsnorb 31972 satfrnmapom 33744 rnmptsn 35737 abrexdom 36120 rncnvepres 36695 imaopab 40585 |
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