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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 6003 | . . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) | |
2 | 1 | dmeqi 5775 | . . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵) |
3 | dmun 5781 | . . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) | |
4 | 2, 3 | eqtri 2846 | . 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
5 | df-rn 5568 | . 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵) | |
6 | df-rn 5568 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 5568 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | uneq12i 4139 | . 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
9 | 4, 5, 8 | 3eqtr4i 2856 | 1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3936 ◡ccnv 5556 dom cdm 5557 ran crn 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: imaundi 6010 imaundir 6011 rnpropg 6081 fun 6542 foun 6635 fpr 6918 sbthlem6 8634 fodomr 8670 brwdom2 9039 ordtval 21799 axlowdimlem13 26742 ex-rn 28221 padct 30457 ffsrn 30467 locfinref 31107 esumrnmpt2 31329 satfrnmapom 32619 noextend 33175 noextendseq 33176 ptrest 34893 rntrclfvOAI 39295 rclexi 39982 rtrclex 39984 rtrclexi 39988 cnvrcl0 39992 rntrcl 39995 dfrtrcl5 39996 dfrcl2 40026 rntrclfv 40084 rnresun 41443 |
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