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Theorem rninc 25384
Description: The range of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
Hypothesis
Ref Expression
rninc.1  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
Assertion
Ref Expression
rninc  |-  ran  C  =  _V
Distinct variable groups:    y, C    x, y
Allowed substitution hint:    C( x)

Proof of Theorem rninc
StepHypRef Expression
1 eqv 3483 . 2  |-  ( ran 
C  =  _V  <->  A. y 
y  e.  ran  C
)
2 ssid 3210 . . . . 5  |-  y  C_  y
3 sseq1 3212 . . . . . . . 8  |-  ( y  =  x  ->  (
y  C_  y  <->  x  C_  y
) )
43equcoms 1666 . . . . . . 7  |-  ( x  =  y  ->  (
y  C_  y  <->  x  C_  y
) )
54biimpd 198 . . . . . 6  |-  ( x  =  y  ->  (
y  C_  y  ->  x 
C_  y ) )
65spimev 1952 . . . . 5  |-  ( y 
C_  y  ->  E. x  x  C_  y )
72, 6ax-mp 8 . . . 4  |-  E. x  x  C_  y
8 abid 2284 . . . 4  |-  ( y  e.  { y  |  E. x  x  C_  y }  <->  E. x  x  C_  y )
97, 8mpbir 200 . . 3  |-  y  e. 
{ y  |  E. x  x  C_  y }
10 rninc.1 . . . . 5  |-  C  =  { <. x ,  y
>.  |  x  C_  y }
1110rneqi 4921 . . . 4  |-  ran  C  =  ran  { <. x ,  y >.  |  x 
C_  y }
12 rnopab 4940 . . . 4  |-  ran  { <. x ,  y >.  |  x  C_  y }  =  { y  |  E. x  x  C_  y }
1311, 12eqtri 2316 . . 3  |-  ran  C  =  { y  |  E. x  x  C_  y }
149, 13eleqtrri 2369 . 2  |-  y  e. 
ran  C
151, 14mpgbir 1540 1  |-  ran  C  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   {copab 4092   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-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-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716
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