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Theorem rnin 4676
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 4674 . . . 4 (AB) = (AB)
21dmeqi 4479 . . 3 dom (AB) = dom (AB)
3 dmin 4486 . . 3 dom (AB) ⊆ (dom A ∩ dom B)
42, 3eqsstri 2969 . 2 dom (AB) ⊆ (dom A ∩ dom B)
5 df-rn 4299 . 2 ran (AB) = dom (AB)
6 df-rn 4299 . . 3 ran A = dom A
7 df-rn 4299 . . 3 ran B = dom B
86, 7ineq12i 3130 . 2 (ran A ∩ ran B) = (dom A ∩ dom B)
94, 5, 83sstr4i 2978 1 ran (AB) ⊆ (ran A ∩ ran B)
Colors of variables: wff set class
Syntax hints:  cin 2910  wss 2911  ccnv 4287  dom cdm 4288  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  inimass  4683
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