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Theorem rnin 4916
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 4914 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4708 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4715 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `' A  i^i  dom  `' B )
42, 3eqsstri 3097 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `' A  i^i  dom  `' B
)
5 df-rn 4518 . 2  |-  ran  ( A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4518 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4518 . . 3  |-  ran  B  =  dom  `' B
86, 7ineq12i 3243 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `' A  i^i  dom  `' B
)
94, 5, 83sstr4i 3106 1  |-  ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3038    C_ wss 3039   `'ccnv 4506   dom cdm 4507   ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  inimass  4923
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