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Theorem rnin 5089
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 5087 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4879 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4885 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `'  A  i^i  dom  `'  B )
42, 3eqsstri 3209 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `'  A  i^i  dom  `'  B
)
5 df-rn 4699 . 2  |-  ran  (  A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4699 . . 3  |-  ran  A  =  dom  `'  A
7 df-rn 4699 . . 3  |-  ran  B  =  dom  `'  B
86, 7ineq12i 3369 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `'  A  i^i  dom  `'  B
)
94, 5, 83sstr4i 3218 1  |-  ran  (  A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3152    C_ wss 3153   `'ccnv 4687   dom cdm 4688   ran crn 4689
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699
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