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Mirrors > Home > MPE Home > Th. List > rnun | Structured version Visualization version GIF version |
Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
rnun | ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvun 6142 | . . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) | |
2 | 1 | dmeqi 5904 | . . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵) |
3 | dmun 5910 | . . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) | |
4 | 2, 3 | eqtri 2760 | . 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
5 | df-rn 5687 | . 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵) | |
6 | df-rn 5687 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5687 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | uneq12i 4161 | . 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
9 | 4, 5, 8 | 3eqtr4i 2770 | 1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3946 ◡ccnv 5675 dom cdm 5676 ran crn 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: imaundi 6149 imaundir 6150 rnpropg 6221 fun 6753 foun 6851 fpr 7151 sbthlem6 9087 fodomr 9127 brwdom2 9567 ordtval 22692 noextend 27166 noextendseq 27167 axlowdimlem13 28209 ex-rn 29690 padct 31939 ffsrn 31949 locfinref 32816 esumrnmpt2 33061 satfrnmapom 34356 ptrest 36482 rntrclfvOAI 41419 tfsconcatrn 42082 rclexi 42356 rtrclex 42358 rtrclexi 42362 cnvrcl0 42366 rntrcl 42369 dfrtrcl5 42370 dfrcl2 42415 rntrclfv 42473 rnresun 43866 |
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