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Theorem rnin 4757
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  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )

Proof of Theorem rnin
StepHypRef Expression
1 cnvin 4755 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4558 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4565 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `' A  i^i  dom  `' B )
42, 3eqsstri 3030 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `' A  i^i  dom  `' B
)
5 df-rn 4376 . 2  |-  ran  ( A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4376 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4376 . . 3  |-  ran  B  =  dom  `' B
86, 7ineq12i 3166 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `' A  i^i  dom  `' B
)
94, 5, 83sstr4i 3039 1  |-  ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 2973    C_ wss 2974   `'ccnv 4364   dom cdm 4365   ran crn 4366
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 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-dm 4375  df-rn 4376
This theorem is referenced by:  inimass  4764
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