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Theorem rnin 4783
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 4781 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 4584 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 4591 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 3038 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 4402 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 4402 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 4402 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 3181 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 3047 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  cin 2981  wss 2982  ccnv 4390  dom cdm 4391  ran crn 4392
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  inimass  4790
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