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Mirrors > Home > ILE Home > Th. List > rnin | GIF version |
Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
rnin | ⊢ ran (A ∩ B) ⊆ (ran A ∩ ran B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvin 4674 | . . . 4 ⊢ ◡(A ∩ B) = (◡A ∩ ◡B) | |
2 | 1 | dmeqi 4479 | . . 3 ⊢ dom ◡(A ∩ B) = dom (◡A ∩ ◡B) |
3 | dmin 4486 | . . 3 ⊢ dom (◡A ∩ ◡B) ⊆ (dom ◡A ∩ dom ◡B) | |
4 | 2, 3 | eqsstri 2969 | . 2 ⊢ dom ◡(A ∩ B) ⊆ (dom ◡A ∩ dom ◡B) |
5 | df-rn 4299 | . 2 ⊢ ran (A ∩ B) = dom ◡(A ∩ B) | |
6 | df-rn 4299 | . . 3 ⊢ ran A = dom ◡A | |
7 | df-rn 4299 | . . 3 ⊢ ran B = dom ◡B | |
8 | 6, 7 | ineq12i 3130 | . 2 ⊢ (ran A ∩ ran B) = (dom ◡A ∩ dom ◡B) |
9 | 4, 5, 8 | 3sstr4i 2978 | 1 ⊢ ran (A ∩ B) ⊆ (ran A ∩ ran B) |
Colors of variables: wff set class |
Syntax hints: ∩ cin 2910 ⊆ wss 2911 ◡ccnv 4287 dom cdm 4288 ran crn 4289 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: inimass 4683 |
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