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Theorem rnin 5043
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 5041 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4833 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4839 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `'  A  i^i  dom  `'  B )
42, 3eqsstri 3150 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `'  A  i^i  dom  `'  B
)
5 df-rn 4645 . 2  |-  ran  (  A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4645 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4645 . . 3  |-  ran  B  =  dom  `'  B
86, 7ineq12i 3310 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `'  A  i^i  dom  `'  B
)
94, 5, 83sstr4i 3159 1  |-  ran  (  A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3093    C_ wss 3094   `'ccnv 4625   dom cdm 4626   ran crn 4627
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-rel 4641  df-cnv 4642  df-dm 4644  df-rn 4645
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