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Theorem rnintintrn 25126
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 3919 . . 3  |-  ( A  =  (/)  ->  |^|_ x  e.  A  ran  B  = 
|^|_ x  e.  (/)  ran  B
)
2 0iin 3960 . . 3  |-  |^|_ x  e.  (/)  ran  B  =  _V
3 ssv 3198 . . . 4  |-  ran  |^|_ x  e.  A  B  C_  _V
4 eqtr 2300 . . . 4  |-  ( (
|^|_ x  e.  A  ran  B  =  |^|_ x  e.  (/)  ran  B  /\  |^|_
x  e.  (/)  ran  B  =  _V )  ->  |^|_ x  e.  A  ran  B  =  _V )
53, 4syl5sseqr 3227 . . 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 5212 . . . . 5  |-  ( A  =/=  (/)  ->  `' |^|_ x  e.  A  B  =  |^|_
x  e.  A  `' B )
87dmeqd 4881 . . . 4  |-  ( A  =/=  (/)  ->  dom  `' |^|_ x  e.  A  B  =  dom  |^|_ x  e.  A  `' B )
9 dmiin 4922 . . . . 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 3212 . . 3  |-  ( A  =/=  (/)  ->  dom  `' |^|_ x  e.  A  B  C_  |^|_
x  e.  A  dom  `' B )
12 df-rn 4700 . . 3  |-  ran  |^|_ x  e.  A  B  =  dom  `' |^|_ x  e.  A  B
13 df-rn 4700 . . . . 5  |-  ran  B  =  dom  `' B
1413a1i 10 . . . 4  |-  ( x  e.  A  ->  ran  B  =  dom  `' B
)
1514iineq2i 3924 . . 3  |-  |^|_ x  e.  A  ran  B  = 
|^|_ x  e.  A  dom  `' B
1611, 12, 153sstr4g 3219 . 2  |-  ( A  =/=  (/)  ->  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B )
176, 16pm2.61ine 2522 1  |-  ran  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  ran  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   |^|_ciin 3906   `'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-iin 3908  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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