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Theorem dfcnv2 27672
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 5384 . 2  |-  Rel  `' R
2 relxp 5119 . . . 4  |-  Rel  ( { x }  X.  ( `' R " { x } ) )
32rgenw 2818 . . 3  |-  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) )
4 reliun 5132 . . 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 3112 . . . . . . . . 9  |-  z  e. 
_V
7 vex 3112 . . . . . . . . 9  |-  y  e. 
_V
86, 7opeldm 5216 . . . . . . . 8  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
dom  `' R )
9 df-rn 5019 . . . . . . . 8  |-  ran  R  =  dom  `' R
108, 9syl6eleqr 2556 . . . . . . 7  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
ran  R )
11 ssel2 3494 . . . . . . 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 5372 . . . . 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 4042 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
1918imaeq2d 5347 . . . 4  |-  ( x  =  z  ->  ( `' R " { x } )  =  ( `' R " { z } ) )
2019opeliunxp2 5151 . . 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 5108 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 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   {csn 4032   <.cop 4038   U_ciun 4332    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-iun 4334  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021
This theorem is referenced by:  gsummpt2co  27931
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