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Theorem dfcnv2 26125
Description: Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
Assertion
Ref Expression
dfcnv2  |-  ( ran 
R  C_  A  ->  `' R  =  U_ x  e.  A  ( {
x }  X.  ( `' R " { x } ) ) )
Distinct variable groups:    x, A    x, R

Proof of Theorem dfcnv2
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5301 . 2  |-  Rel  `' R
2 relxp 5042 . . . 4  |-  Rel  ( { x }  X.  ( `' R " { x } ) )
32rgenw 2888 . . 3  |-  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) )
4 reliun 5055 . . 3  |-  ( Rel  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) ) )
53, 4mpbir 209 . 2  |-  Rel  U_ x  e.  A  ( {
x }  X.  ( `' R " { x } ) )
6 vex 3068 . . . . . . . . 9  |-  z  e. 
_V
7 vex 3068 . . . . . . . . 9  |-  y  e. 
_V
86, 7opeldm 5138 . . . . . . . 8  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
dom  `' R )
9 df-rn 4946 . . . . . . . 8  |-  ran  R  =  dom  `' R
108, 9syl6eleqr 2548 . . . . . . 7  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
ran  R )
11 ssel2 3446 . . . . . . 7  |-  ( ( ran  R  C_  A  /\  z  e.  ran  R )  ->  z  e.  A )
1210, 11sylan2 474 . . . . . 6  |-  ( ( ran  R  C_  A  /\  <. z ,  y
>.  e.  `' R )  ->  z  e.  A
)
1312ex 434 . . . . 5  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  ->  z  e.  A ) )
1413pm4.71rd 635 . . . 4  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) ) )
156, 7elimasn 5289 . . . . 5  |-  ( y  e.  ( `' R " { z } )  <->  <. z ,  y >.  e.  `' R )
1615anbi2i 694 . . . 4  |-  ( ( z  e.  A  /\  y  e.  ( `' R " { z } ) )  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) )
1714, 16syl6bbr 263 . . 3  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) ) )
18 sneq 3982 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
1918imaeq2d 5264 . . . 4  |-  ( x  =  z  ->  ( `' R " { x } )  =  ( `' R " { z } ) )
2019opeliunxp2 5073 . . 3  |-  ( <.
z ,  y >.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) )
2117, 20syl6bbr 263 . 2  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  <. z ,  y
>.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) ) ) )
221, 5, 21eqrelrdv 5031 1  |-  ( ran 
R  C_  A  ->  `' R  =  U_ x  e.  A  ( {
x }  X.  ( `' R " { x } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793    C_ wss 3423   {csn 3972   <.cop 3978   U_ciun 4266    X. cxp 4933   `'ccnv 4934   dom cdm 4935   ran crn 4936   "cima 4938   Rel wrel 4940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-iun 4268  df-br 4388  df-opab 4446  df-xp 4941  df-rel 4942  df-cnv 4943  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948
This theorem is referenced by:  gsummpt2co  26380
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