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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 6132 | . . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) | |
2 | 1 | dmeqi 5897 | . . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵) |
3 | dmun 5903 | . . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) | |
4 | 2, 3 | eqtri 2760 | . 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
5 | df-rn 5681 | . 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵) | |
6 | df-rn 5681 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5681 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | uneq12i 4158 | . 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 3943 ◡ccnv 5669 dom cdm 5670 ran crn 5671 |
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 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-cnv 5678 df-dm 5680 df-rn 5681 |
This theorem is referenced by: imaundi 6139 imaundir 6140 rnpropg 6211 fun 6741 foun 6839 fpr 7137 sbthlem6 9073 fodomr 9113 brwdom2 9552 ordtval 22624 noextend 27098 noextendseq 27099 axlowdimlem13 28141 ex-rn 29622 padct 31879 ffsrn 31889 locfinref 32716 esumrnmpt2 32961 satfrnmapom 34256 ptrest 36355 rntrclfvOAI 41264 tfsconcatrn 41927 rclexi 42201 rtrclex 42203 rtrclexi 42207 cnvrcl0 42211 rntrcl 42214 dfrtrcl5 42215 dfrcl2 42260 rntrclfv 42318 rnresun 43711 |
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