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Theorem rnintintrn 25229
Description: The range of an intersection is a part of the intersection of the ranges. (The case  A  =  (/) works as well, the intersection gives  _V). (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
rnintintrn  |-  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem rnintintrn
StepHypRef Expression
1 iineq1 3935 . . 3  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ran  B  = 
|^|_ x  e.  (/)  ran  B
)
2 0iin 3976 . . 3  |-  |^|_ x  e.  (/)  ran  B  =  _V
3 ssv 3211 . . . 4  |-  ran  |^|_ x  e.  A  B  C_  _V
4 eqtr 2313 . . . 4  |-  ( (
|^|_ x  e.  A  ran  B  =  |^|_ x  e.  (/)  ran  B  /\  |^|_
x  e.  (/)  ran  B  =  _V )  ->  |^|_ x  e.  A  ran  B  =  _V )
53, 4syl5sseqr 3240 . . 3  |-  ( (
|^|_ x  e.  A  ran  B  =  |^|_ x  e.  (/)  ran  B  /\  |^|_
x  e.  (/)  ran  B  =  _V )  ->  ran  |^|_
x  e.  A  B  C_ 
|^|_ x  e.  A  ran  B )
61, 2, 5sylancl 643 . 2  |-  ( A  =  (/)  ->  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B )
7 cnviin 5228 . . . . 5  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
87dmeqd 4897 . . . 4  |-  ( A  =/=  (/)  ->  dom  `' |^|_ x  e.  A  B  =  dom  |^|_ x  e.  A  `' B )
9 dmiin 4938 . . . . 5  |-  dom  |^|_ x  e.  A  `' B  C_ 
|^|_ x  e.  A  dom  `' B
109a1i 10 . . . 4  |-  ( A  =/=  (/)  ->  dom  |^|_ x  e.  A  `' B  C_ 
|^|_ x  e.  A  dom  `' B )
118, 10eqsstrd 3225 . . 3  |-  ( A  =/=  (/)  ->  dom  `' |^|_ x  e.  A  B  C_  |^|_
x  e.  A  dom  `' B )
12 df-rn 4716 . . 3  |-  ran  |^|_ x  e.  A  B  =  dom  `' |^|_ x  e.  A  B
13 df-rn 4716 . . . . 5  |-  ran  B  =  dom  `' B
1413a1i 10 . . . 4  |-  ( x  e.  A  ->  ran  B  =  dom  `' B
)
1514iineq2i 3940 . . 3  |-  |^|_ x  e.  A  ran  B  = 
|^|_ x  e.  A  dom  `' B
1611, 12, 153sstr4g 3232 . 2  |-  ( A  =/=  (/)  ->  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B )
176, 16pm2.61ine 2535 1  |-  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   |^|_ciin 3922   `'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-iin 3924  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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