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Theorem elrnmpt2res 6774
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 2626 . . . . . 6 |- ( z = D -> ( z = C <-> D = C ) )
21anbi1d 741 . . . . 5 |- ( z = D -> ( ( z = C /\ x R y ) <-> ( D = C /\ x R y ) ) )
32anbi2d 740 . . . 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 1852 . . 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 838 . . . . . . . . . 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 838 . . . . . . . . . . . 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 466 . . . . . . . . . . . . . 14 |- ( ( z = C /\ p e. R ) <-> ( p e. R /\ z = C ) )
8 eleq1 2689 . . . . . . . . . . . . . . . 16 |- ( p = <. x , y >. -> ( p e. R <-> <. x , y >. e. R ) )
9 df-br 4654 . . . . . . . . . . . . . . . 16 |- ( x R y <-> <. x , y >. e. R )
108, 9syl6bbr 278 . . . . . . . . . . . . . . 15 |- ( p = <. x , y >. -> ( p e. R <-> x R y ) )
1110anbi2d 740 . . . . . . . . . . . . . 14 |- ( p = <. x , y >. -> ( ( z = C /\ p e. R ) <-> ( z = C /\ x R y ) ) )
127, 11syl5bbr 274 . . . . . . . . . . . . 13 |- ( p = <. x , y >. -> ( ( p e. R /\ z = C ) <-> ( z = C /\ x R y ) ) )
1312anbi2d 740 . . . . . . . . . . . 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 272 . . . . . . . . . . 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 669 . . . . . . . . . 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 264 . . . . . . . . 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 1775 . . . . . . . 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 1920 . . . . . . . 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 266 . . . . . . 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 4717 . . . . . 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 6701 . . . . . 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 6655 . . . . . . . . 9 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
24 dfoprab2 6701 . . . . . . . . 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 2648 . . . . . . . 8 |- F = { <. p , z >. | E. x E. y ( p = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ z = C ) ) }
2625reseq1i 5392 . . . . . . 7 |- ( F |` R ) = ( { <. p , z >. | E. x E. y ( p = <. x , y >. /\ ( ( x e. A /\ y e. B ) /\ z = C ) ) } |` R )
27 resopab 5446 . . . . . . 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 2644 . . . . . 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 2655 . . . . 5 |- ( F |` R ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ( z = C /\ x R y ) ) }
3029rneqi 5352 . . . 4 |- ran ( F |` R ) = ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ( z = C /\ x R y ) ) }
31 rnoprab 6743 . . . 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 2644 . . 3 |- ran ( F |` R ) = { z | E. x E. y ( ( x e. A /\ y e. B ) /\ ( z = C /\ x R y ) ) }
334, 32elab2g 3353 . 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 3061 . 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 278 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 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   <.cop 4183   class class class wbr 4653   {copab 4712   ran crn 5115   |` cres 5116   {coprab 6651   |-> cmpt2 6652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-oprab 6654  df-mpt2 6655
This theorem is referenced by: (None)
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