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Theorem dfcnv2 28225
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 5169 . 2  |-  Rel  `' R
2 relxp 4904 . . . 4  |-  Rel  ( { x }  X.  ( `' R " { x } ) )
32rgenw 2726 . . 3  |-  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) )
4 reliun 4916 . . 3  |-  ( Rel  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  A. x  e.  A  Rel  ( { x }  X.  ( `' R " { x } ) ) )
53, 4mpbir 212 . 2  |-  Rel  U_ x  e.  A  ( {
x }  X.  ( `' R " { x } ) )
6 vex 3025 . . . . . . . . 9  |-  z  e. 
_V
7 vex 3025 . . . . . . . . 9  |-  y  e. 
_V
86, 7opeldm 5000 . . . . . . . 8  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
dom  `' R )
9 df-rn 4807 . . . . . . . 8  |-  ran  R  =  dom  `' R
108, 9syl6eleqr 2517 . . . . . . 7  |-  ( <.
z ,  y >.  e.  `' R  ->  z  e. 
ran  R )
11 ssel2 3402 . . . . . . 7  |-  ( ( ran  R  C_  A  /\  z  e.  ran  R )  ->  z  e.  A )
1210, 11sylan2 476 . . . . . 6  |-  ( ( ran  R  C_  A  /\  <. z ,  y
>.  e.  `' R )  ->  z  e.  A
)
1312ex 435 . . . . 5  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  ->  z  e.  A ) )
1413pm4.71rd 639 . . . 4  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) ) )
156, 7elimasn 5155 . . . . 5  |-  ( y  e.  ( `' R " { z } )  <->  <. z ,  y >.  e.  `' R )
1615anbi2i 698 . . . 4  |-  ( ( z  e.  A  /\  y  e.  ( `' R " { z } ) )  <->  ( z  e.  A  /\  <. z ,  y >.  e.  `' R ) )
1714, 16syl6bbr 266 . . 3  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) ) )
18 sneq 3951 . . . . 5  |-  ( x  =  z  ->  { x }  =  { z } )
1918imaeq2d 5130 . . . 4  |-  ( x  =  z  ->  ( `' R " { x } )  =  ( `' R " { z } ) )
2019opeliunxp2 4935 . . 3  |-  ( <.
z ,  y >.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) )  <->  ( z  e.  A  /\  y  e.  ( `' R " { z } ) ) )
2117, 20syl6bbr 266 . 2  |-  ( ran 
R  C_  A  ->  (
<. z ,  y >.  e.  `' R  <->  <. z ,  y
>.  e.  U_ x  e.  A  ( { x }  X.  ( `' R " { x } ) ) ) )
221, 5, 21eqrelrdv 4893 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 370    = wceq 1437    e. wcel 1872   A.wral 2714    C_ wss 3379   {csn 3941   <.cop 3947   U_ciun 4242    X. cxp 4794   `'ccnv 4795   dom cdm 4796   ran crn 4797   "cima 4799   Rel wrel 4801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-iun 4244  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809
This theorem is referenced by:  gsummpt2co  28494
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