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Theorem rnin 5106
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 5104 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4896 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4902 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `' A  i^i  dom  `' B )
42, 3eqsstri 3221 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `' A  i^i  dom  `' B
)
5 df-rn 4716 . 2  |-  ran  ( A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4716 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4716 . . 3  |-  ran  B  =  dom  `' B
86, 7ineq12i 3381 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `' A  i^i  dom  `' B
)
94, 5, 83sstr4i 3230 1  |-  ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3164    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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