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Theorem rnin 6153
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 6151 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 5907 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 5914 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 4011 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 5689 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 5689 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 5689 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 4208 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 4020 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  cin 3943  wss 3944  ccnv 5677  dom cdm 5678  ran crn 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  inimass  6161  restutop  24186
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