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Theorem ssrnres 4706
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres 
C_  ran  C  |`  ran  C  i^i  X.

Proof of Theorem ssrnres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3152 . . . . 5  C  i^i  X.  C_  X.
2 rnss 4507 . . . . 5  C  i^i  X. 
C_  X.  ran  C  i^i  X.  C_  ran  X.
31, 2ax-mp 7 . . . 4  ran  C  i^i  X.  C_  ran  X.
4 rnxpss 4697 . . . 4  ran  X.  C_
53, 4sstri 2948 . . 3  ran  C  i^i  X.  C_
6 eqss 2954 . . 3  ran  C  i^i  X.  ran  C  i^i  X. 
C_  C_ 
ran  C  i^i  X.
75, 6mpbiran 846 . 2  ran  C  i^i  X.  C_  ran  C  i^i  X.
8 ssid 2958 . . . . . . . 8  C_
9 ssv 2959 . . . . . . . 8  C_  _V
10 xpss12 4388 . . . . . . . 8  C_  C_  _V  X.  C_  X.  _V
118, 9, 10mp2an 402 . . . . . . 7  X.  C_  X.  _V
12 sslin 3157 . . . . . . 7  X. 
C_  X.  _V  C  i^i  X.  C_  C  i^i  X.  _V
1311, 12ax-mp 7 . . . . . 6  C  i^i  X.  C_  C  i^i  X.  _V
14 df-res 4300 . . . . . 6  C  |`  C  i^i  X.  _V
1513, 14sseqtr4i 2972 . . . . 5  C  i^i  X.  C_  C  |`
16 rnss 4507 . . . . 5  C  i^i  X. 
C_  C  |`  ran  C  i^i  X.  C_  ran  C  |`
1715, 16ax-mp 7 . . . 4  ran  C  i^i  X.  C_  ran  C  |`
18 sstr 2947 . . . 4  C_  ran  C  i^i  X.  ran  C  i^i  X. 
C_  ran  C  |`  C_  ran  C  |`
1917, 18mpan2 401 . . 3 
C_  ran  C  i^i  X.  C_  ran  C  |`
20 ssel 2933 . . . . . . 7 
C_  ran  C  |`  ran  C  |`
21 vex 2554 . . . . . . . 8 
_V
2221elrn2 4519 . . . . . . 7  ran  C  |`  <. ,  >.  C  |`
2320, 22syl6ib 150 . . . . . 6 
C_  ran  C  |`  <. , 
>.  C  |`
2423ancrd 309 . . . . 5 
C_  ran  C  |`  <. ,  >.  C  |`
2521elrn2 4519 . . . . . 6  ran  C  i^i  X.  <. ,  >.  C  i^i  X.
26 elin 3120 . . . . . . . 8  <. ,  >.  C  i^i  X.  <. , 
>.  C  <. ,  >.  X.
27 opelxp 4317 . . . . . . . . 9  <. ,  >.  X.
2827anbi2i 430 . . . . . . . 8 
<. ,  >.  C  <. ,  >.  X.  <. , 
>.  C
2921opelres 4560 . . . . . . . . . 10  <. ,  >.  C  |`  <. , 
>.  C
3029anbi1i 431 . . . . . . . . 9 
<. ,  >.  C  |`  <. ,  >.  C
31 anass 381 . . . . . . . . 9  <. , 
>.  C  <. ,  >.  C
3230, 31bitr2i 174 . . . . . . . 8 
<. ,  >.  C  <. ,  >.  C  |`
3326, 28, 323bitri 195 . . . . . . 7  <. ,  >.  C  i^i  X.  <. , 
>.  C  |`
3433exbii 1493 . . . . . 6  <. , 
>.  C  i^i  X.  <. , 
>.  C  |`
35 19.41v 1779 . . . . . 6  <. , 
>.  C  |`  <. , 
>.  C  |`
3625, 34, 353bitri 195 . . . . 5  ran  C  i^i  X.  <. , 
>.  C  |`
3724, 36syl6ibr 151 . . . 4 
C_  ran  C  |`  ran  C  i^i  X.
3837ssrdv 2945 . . 3 
C_  ran  C  |`  C_ 
ran  C  i^i  X.
3919, 38impbii 117 . 2 
C_  ran  C  i^i  X.  C_  ran  C  |`
407, 39bitr2i 174 1 
C_  ran  C  |`  ran  C  i^i  X.
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551    i^i cin 2910    C_ wss 2911   <.cop 3370    X. cxp 4286   ran crn 4289    |` cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by:  rninxp  4707
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