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Theorem rnin 5461
 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 5459 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 5247 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 5254 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 3598 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 5049 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 5049 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 5049 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 3774 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 3607 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3539   ⊆ wss 3540  ◡ccnv 5037  dom cdm 5038  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  inimass  5468  restutop  21851
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