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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 6143 | . . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) | |
2 | 1 | dmeqi 5905 | . . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵) |
3 | dmun 5911 | . . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) | |
4 | 2, 3 | eqtri 2761 | . 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
5 | df-rn 5688 | . 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵) | |
6 | df-rn 5688 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5688 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | uneq12i 4162 | . 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
9 | 4, 5, 8 | 3eqtr4i 2771 | 1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3947 ◡ccnv 5676 dom cdm 5677 ran crn 5678 |
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-12 2172 ax-ext 2704 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-cnv 5685 df-dm 5687 df-rn 5688 |
This theorem is referenced by: imaundi 6150 imaundir 6151 rnpropg 6222 fun 6754 foun 6852 fpr 7152 sbthlem6 9088 fodomr 9128 brwdom2 9568 ordtval 22693 noextend 27169 noextendseq 27170 axlowdimlem13 28212 ex-rn 29693 padct 31944 ffsrn 31954 locfinref 32821 esumrnmpt2 33066 satfrnmapom 34361 ptrest 36487 rntrclfvOAI 41429 tfsconcatrn 42092 rclexi 42366 rtrclex 42368 rtrclexi 42372 cnvrcl0 42376 rntrcl 42379 dfrtrcl5 42380 dfrcl2 42425 rntrclfv 42483 rnresun 43876 |
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