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Theorem rnin 5090
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 5088 . . . 4  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
21dmeqi 4880 . . 3  |-  dom  `' ( A  i^i  B )  =  dom  ( `' A  i^i  `' B
)
3 dmin 4886 . . 3  |-  dom  ( `' A  i^i  `' B
)  C_  ( dom  `' A  i^i  dom  `' B )
42, 3eqsstri 3208 . 2  |-  dom  `' ( A  i^i  B ) 
C_  ( dom  `' A  i^i  dom  `' B
)
5 df-rn 4700 . 2  |-  ran  ( A  i^i  B )  =  dom  `' ( A  i^i  B )
6 df-rn 4700 . . 3  |-  ran  A  =  dom  `' A
7 df-rn 4700 . . 3  |-  ran  B  =  dom  `' B
86, 7ineq12i 3368 . 2  |-  ( ran 
A  i^i  ran  B )  =  ( dom  `' A  i^i  dom  `' B
)
94, 5, 83sstr4i 3217 1  |-  ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
Colors of variables: wff set class
Syntax hints:    i^i cin 3151    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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