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Theorem dfcnv2 28328
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 5226 . 2  |-  Rel  `' R
2 relxp 4961 . . . 4  |-  Rel  ( { x }  X.  ( `' R " { x } ) )
32rgenw 2761 . . 3  |-  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) )
4 reliun 4973 . . 3  |-  ( Rel  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) ) )
53, 4mpbir 214 . 2  |-  Rel  U_ x  e.  A  ( {
x }  X.  ( `' R " { x } ) )
6 vex 3060 . . . . . . . . 9  |-  z  e. 
_V
7 vex 3060 . . . . . . . . 9  |-  y  e. 
_V
86, 7opeldm 5057 . . . . . . . 8  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
dom  `' R )
9 df-rn 4864 . . . . . . . 8  |-  ran  R  =  dom  `' R
108, 9syl6eleqr 2551 . . . . . . 7  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
ran  R )
11 ssel2 3439 . . . . . . 7  |-  ( ( ran  R  C_  A  /\  z  e.  ran  R )  ->  z  e.  A )
1210, 11sylan2 481 . . . . . 6  |-  ( ( ran  R  C_  A  /\  <. z ,  y
>.  e.  `' R )  ->  z  e.  A
)
1312ex 440 . . . . 5  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  ->  z  e.  A ) )
1413pm4.71rd 645 . . . 4  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) ) )
156, 7elimasn 5212 . . . . 5  |-  ( y  e.  ( `' R " { z } )  <->  <. z ,  y >.  e.  `' R )
1615anbi2i 705 . . . 4  |-  ( ( z  e.  A  /\  y  e.  ( `' R " { z } ) )  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) )
1714, 16syl6bbr 271 . . 3  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) ) )
18 sneq 3990 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
1918imaeq2d 5187 . . . 4  |-  ( x  =  z  ->  ( `' R " { x } )  =  ( `' R " { z } ) )
2019opeliunxp2 4992 . . 3  |-  ( <.
z ,  y >.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) )
2117, 20syl6bbr 271 . 2  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  <. z ,  y
>.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) ) ) )
221, 5, 21eqrelrdv 4950 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 375    = wceq 1455    e. wcel 1898   A.wral 2749    C_ wss 3416   {csn 3980   <.cop 3986   U_ciun 4292    X. cxp 4851   `'ccnv 4852   dom cdm 4853   ran crn 4854   "cima 4856   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-iun 4294  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by:  gsummpt2co  28592
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