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Theorem rnin 4828
Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
rnin ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Proof of Theorem rnin
StepHypRef Expression
1 cnvin 4826 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4625 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 4632 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 3054 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 4439 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4439 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4439 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 3197 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 3063 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  cin 2996  wss 2997  ccnv 4427  dom cdm 4428  ran crn 4429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  inimass  4835
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