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Theorem rnin 6045
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 6043 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 5808 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 5815 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 3956 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 5597 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 5597 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 5597 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 4146 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 3965 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  cin 3887  wss 3888  ccnv 5585  dom cdm 5586  ran crn 5587
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5076  df-opab 5138  df-xp 5592  df-rel 5593  df-cnv 5594  df-dm 5596  df-rn 5597
This theorem is referenced by:  inimass  6053  restutop  23378
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