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Mirrors > Home > MPE Home > Th. List > dmun | Structured version Visualization version GIF version |
Description: The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmun | ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unab 4297 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} | |
2 | brun 5194 | . . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
3 | 2 | exbii 1843 | . . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) |
4 | 19.43 1878 | . . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)) | |
5 | 3, 4 | bitr2i 275 | . . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥) |
6 | 5 | abbii 2796 | . . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} |
7 | 1, 6 | eqtri 2754 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} |
8 | df-dm 5682 | . . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} | |
9 | df-dm 5682 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥} | |
10 | 8, 9 | uneq12i 4158 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) |
11 | df-dm 5682 | . 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} | |
12 | 7, 10, 11 | 3eqtr4ri 2765 | 1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 845 = wceq 1534 ∃wex 1774 {cab 2703 ∪ cun 3944 class class class wbr 5143 dom cdm 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3951 df-br 5144 df-dm 5682 |
This theorem is referenced by: rnun 6147 dmpropg 6216 dmtpop 6219 fntpg 6608 fnun 6663 frrlem14 8303 wfrlem13OLD 8340 wfrlem16OLD 8343 tfrlem10 8406 sbthlem5 9114 fodomr 9155 fodomfir 9360 axdc3lem4 10484 hashfun 14446 s4dom 14920 dmtrclfv 15015 strleun 17151 setsdm 17164 estrreslem2 18154 mvdco 19436 gsumzaddlem 19912 cnfldfunALT 21351 cnfldfunALTOLD 21364 cnfldfunALTOLDOLD 21365 noextend 27690 noextendseq 27691 nosupbday 27729 nosupbnd1 27738 nosupbnd2 27740 noinfbday 27744 noinfbnd1 27753 noinfbnd2 27755 noetasuplem4 27760 noetainflem4 27764 uhgrun 29004 upgrun 29048 umgrun 29050 vtxdun 29412 wlkp1 29612 eupthp1 30143 bnj1416 34894 fineqvac 34943 satfdm 35207 fmlasuc0 35222 fixun 35743 rclexi 43316 rtrclex 43318 rtrclexi 43322 cnvrcl0 43326 dmtrcl 43328 dfrtrcl5 43330 dfrcl2 43375 dmtrclfvRP 43431 |
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