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Theorem rnin 5682
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 5680 . . . 4 (𝐴𝐵) = (𝐴𝐵)
21dmeqi 5462 . . 3 dom (𝐴𝐵) = dom (𝐴𝐵)
3 dmin 5469 . . 3 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
42, 3eqsstri 3784 . 2 dom (𝐴𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
5 df-rn 5261 . 2 ran (𝐴𝐵) = dom (𝐴𝐵)
6 df-rn 5261 . . 3 ran 𝐴 = dom 𝐴
7 df-rn 5261 . . 3 ran 𝐵 = dom 𝐵
86, 7ineq12i 3963 . 2 (ran 𝐴 ∩ ran 𝐵) = (dom 𝐴 ∩ dom 𝐵)
94, 5, 83sstr4i 3793 1 ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  cin 3722  wss 3723  ccnv 5249  dom cdm 5250  ran crn 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  inimass  5689  restutop  22261
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