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Mirrors > Home > ILE Home > Th. List > rnun | 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 4984 | . . . 4 ⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵) | |
2 | 1 | dmeqi 4780 | . . 3 ⊢ dom ◡(𝐴 ∪ 𝐵) = dom (◡𝐴 ∪ ◡𝐵) |
3 | dmun 4786 | . . 3 ⊢ dom (◡𝐴 ∪ ◡𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) | |
4 | 2, 3 | eqtri 2175 | . 2 ⊢ dom ◡(𝐴 ∪ 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
5 | df-rn 4590 | . 2 ⊢ ran (𝐴 ∪ 𝐵) = dom ◡(𝐴 ∪ 𝐵) | |
6 | df-rn 4590 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 4590 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | uneq12i 3255 | . 2 ⊢ (ran 𝐴 ∪ ran 𝐵) = (dom ◡𝐴 ∪ dom ◡𝐵) |
9 | 4, 5, 8 | 3eqtr4i 2185 | 1 ⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∪ cun 3096 ◡ccnv 4578 dom cdm 4579 ran crn 4580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-v 2711 df-un 3102 df-in 3104 df-ss 3111 df-sn 3562 df-pr 3563 df-op 3565 df-br 3962 df-opab 4022 df-cnv 4587 df-dm 4589 df-rn 4590 |
This theorem is referenced by: imaundi 4991 imaundir 4992 rnpropg 5058 fun 5335 foun 5426 fpr 5642 fprg 5643 sbthlemi6 6895 exmidfodomrlemim 7115 |
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