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Theorem elrnmpt2res 6424
Description: Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypothesis
Ref Expression
rngop.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
elrnmpt2res  |-  ( D  e.  V  ->  ( D  e.  ran  ( F  |`  R )  <->  E. x  e.  A  E. y  e.  B  ( D  =  C  /\  x R y ) ) )
Distinct variable groups:    y, A    x, y, D    x, R, y
Allowed substitution hints:    A( x)    B( x, y)    C( x, y)    F( x, y)    V( x, y)

Proof of Theorem elrnmpt2res
Dummy variables  z  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2433 . . . . . 6  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
21anbi1d 709 . . . . 5  |-  ( z  =  D  ->  (
( z  =  C  /\  x R y )  <->  ( D  =  C  /\  x R y ) ) )
32anbi2d 708 . . . 4  |-  ( z  =  D  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) )  <-> 
( ( x  e.  A  /\  y  e.  B )  /\  ( D  =  C  /\  x R y ) ) ) )
432exbidv 1763 . . 3  |-  ( z  =  D  ->  ( E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) )  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B
)  /\  ( D  =  C  /\  x R y ) ) ) )
5 an12 804 . . . . . . . . . 10  |-  ( ( p  e.  R  /\  ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) )  <-> 
( p  =  <. x ,  y >.  /\  (
p  e.  R  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) ) )
6 an12 804 . . . . . . . . . . . 12  |-  ( ( p  e.  R  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ( p  e.  R  /\  z  =  C
) ) )
7 ancom 451 . . . . . . . . . . . . . 14  |-  ( ( z  =  C  /\  p  e.  R )  <->  ( p  e.  R  /\  z  =  C )
)
8 eleq1 2501 . . . . . . . . . . . . . . . 16  |-  ( p  =  <. x ,  y
>.  ->  ( p  e.  R  <->  <. x ,  y
>.  e.  R ) )
9 df-br 4427 . . . . . . . . . . . . . . . 16  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
108, 9syl6bbr 266 . . . . . . . . . . . . . . 15  |-  ( p  =  <. x ,  y
>.  ->  ( p  e.  R  <->  x R y ) )
1110anbi2d 708 . . . . . . . . . . . . . 14  |-  ( p  =  <. x ,  y
>.  ->  ( ( z  =  C  /\  p  e.  R )  <->  ( z  =  C  /\  x R y ) ) )
127, 11syl5bbr 262 . . . . . . . . . . . . 13  |-  ( p  =  <. x ,  y
>.  ->  ( ( p  e.  R  /\  z  =  C )  <->  ( z  =  C  /\  x R y ) ) )
1312anbi2d 708 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  ( p  e.  R  /\  z  =  C
) )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ( z  =  C  /\  x R y ) ) ) )
146, 13syl5bb 260 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( ( p  e.  R  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ( z  =  C  /\  x R y ) ) ) )
1514pm5.32i 641 . . . . . . . . . 10  |-  ( ( p  =  <. x ,  y >.  /\  (
p  e.  R  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) )  <-> 
( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) ) ) )
165, 15bitri 252 . . . . . . . . 9  |-  ( ( p  e.  R  /\  ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) )  <-> 
( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) ) ) )
17162exbii 1715 . . . . . . . 8  |-  ( E. x E. y ( p  e.  R  /\  ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) )  <->  E. x E. y ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) ) ) )
18 19.42vv 1828 . . . . . . . 8  |-  ( E. x E. y ( p  e.  R  /\  ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) )  <-> 
( p  e.  R  /\  E. x E. y
( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) ) )
1917, 18bitr3i 254 . . . . . . 7  |-  ( E. x E. y ( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) ) )  <->  ( p  e.  R  /\  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) ) )
2019opabbii 4490 . . . . . 6  |-  { <. p ,  z >.  |  E. x E. y ( p  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) ) ) }  =  { <. p ,  z >.  |  ( p  e.  R  /\  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) ) }
21 dfoprab2 6351 . . . . . 6  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  =  C  /\  x R y ) ) }  =  { <. p ,  z >.  |  E. x E. y ( p  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) ) ) }
22 rngop.1 . . . . . . . . 9  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
23 df-mpt2 6310 . . . . . . . . 9  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
24 dfoprab2 6351 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) }  =  { <. p ,  z
>.  |  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) }
2522, 23, 243eqtri 2462 . . . . . . . 8  |-  F  =  { <. p ,  z
>.  |  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) }
2625reseq1i 5121 . . . . . . 7  |-  ( F  |`  R )  =  ( { <. p ,  z
>.  |  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) }  |`  R )
27 resopab 5171 . . . . . . 7  |-  ( {
<. p ,  z >.  |  E. x E. y
( p  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C ) ) }  |`  R )  =  { <. p ,  z >.  |  ( p  e.  R  /\  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) ) }
2826, 27eqtri 2458 . . . . . 6  |-  ( F  |`  R )  =  { <. p ,  z >.  |  ( p  e.  R  /\  E. x E. y ( p  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) ) ) }
2920, 21, 283eqtr4ri 2469 . . . . 5  |-  ( F  |`  R )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) ) }
3029rneqi 5081 . . . 4  |-  ran  ( F  |`  R )  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ( z  =  C  /\  x R y ) ) }
31 rnoprab 6393 . . . 4  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) ) }  =  { z  |  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  (
z  =  C  /\  x R y ) ) }
3230, 31eqtri 2458 . . 3  |-  ran  ( F  |`  R )  =  { z  |  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ( z  =  C  /\  x R y ) ) }
334, 32elab2g 3226 . 2  |-  ( D  e.  V  ->  ( D  e.  ran  ( F  |`  R )  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ( D  =  C  /\  x R y ) ) ) )
34 r2ex 2958 . 2  |-  ( E. x  e.  A  E. y  e.  B  ( D  =  C  /\  x R y )  <->  E. x E. y ( ( x  e.  A  /\  y  e.  B )  /\  ( D  =  C  /\  x R y ) ) )
3533, 34syl6bbr 266 1  |-  ( D  e.  V  ->  ( D  e.  ran  ( F  |`  R )  <->  E. x  e.  A  E. y  e.  B  ( D  =  C  /\  x R y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   E.wrex 2783   <.cop 4008   class class class wbr 4426   {copab 4483   ran crn 4855    |` cres 4856   {coprab 6306    |-> cmpt2 6307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-oprab 6309  df-mpt2 6310
This theorem is referenced by: (None)
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