Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4269 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} | |
2 | brun 5109 | . . . . . 6 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) | |
3 | 2 | exbii 1844 | . . . . 5 ⊢ (∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥 ↔ ∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥)) |
4 | 19.43 1879 | . . . . 5 ⊢ (∃𝑥(𝑦𝐴𝑥 ∨ 𝑦𝐵𝑥) ↔ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)) | |
5 | 3, 4 | bitr2i 278 | . . . 4 ⊢ ((∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥) ↔ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥) |
6 | 5 | abbii 2886 | . . 3 ⊢ {𝑦 ∣ (∃𝑥 𝑦𝐴𝑥 ∨ ∃𝑥 𝑦𝐵𝑥)} = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} |
7 | 1, 6 | eqtri 2844 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} |
8 | df-dm 5559 | . . 3 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} | |
9 | df-dm 5559 | . . 3 ⊢ dom 𝐵 = {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥} | |
10 | 8, 9 | uneq12i 4136 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐵) = ({𝑦 ∣ ∃𝑥 𝑦𝐴𝑥} ∪ {𝑦 ∣ ∃𝑥 𝑦𝐵𝑥}) |
11 | df-dm 5559 | . 2 ⊢ dom (𝐴 ∪ 𝐵) = {𝑦 ∣ ∃𝑥 𝑦(𝐴 ∪ 𝐵)𝑥} | |
12 | 7, 10, 11 | 3eqtr4ri 2855 | 1 ⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1533 ∃wex 1776 {cab 2799 ∪ cun 3933 class class class wbr 5058 dom cdm 5549 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-br 5059 df-dm 5559 |
This theorem is referenced by: rnun 5998 dmpropg 6066 dmtpop 6069 fntpg 6408 fnun 6457 wfrlem13 7961 wfrlem16 7964 tfrlem10 8017 sbthlem5 8625 fodomr 8662 axdc3lem4 9869 hashfun 13792 s4dom 14275 dmtrclfv 14372 setsdm 16511 strleun 16585 estrreslem2 17382 mvdco 18567 gsumzaddlem 19035 cnfldfun 20551 uhgrun 26853 upgrun 26897 umgrun 26899 vtxdun 27257 wlkp1 27457 eupthp1 27989 bnj1416 32306 satfdm 32611 fmlasuc0 32626 frrlem14 33131 noextend 33168 noextendseq 33169 nosupbday 33200 nosupbnd1 33209 nosupbnd2 33211 noetalem3 33214 noetalem4 33215 fixun 33365 rclexi 39968 rtrclex 39970 rtrclexi 39974 cnvrcl0 39978 dmtrcl 39980 dfrtrcl5 39982 dfrcl2 40012 dmtrclfvRP 40068 |
Copyright terms: Public domain | W3C validator |