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Theorem oprabex2gpop 25036
Description: Existence of an operation class abstraction. (A version of mptex 5746 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
Assertion
Ref Expression
oprabex2gpop  |-  ( ( R  e.  B  /\  Rel  R )  ->  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  R  /\  z  =  A ) }  e.  _V )
Distinct variable groups:    z, A    x, B, y, z    x, R, y, z
Allowed substitution hints:    A( x, y)

Proof of Theorem oprabex2gpop
StepHypRef Expression
1 relssdmrn 5193 . . . . . 6  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
21sseld 3179 . . . . 5  |-  ( Rel 
R  ->  ( <. x ,  y >.  e.  R  -> 
<. x ,  y >.  e.  ( dom  R  X.  ran  R ) ) )
3 opelxp 4719 . . . . 5  |-  ( <.
x ,  y >.  e.  ( dom  R  X.  ran  R )  <->  ( x  e.  dom  R  /\  y  e.  ran  R ) )
42, 3syl6ib 217 . . . 4  |-  ( Rel 
R  ->  ( <. x ,  y >.  e.  R  ->  ( x  e.  dom  R  /\  y  e.  ran  R ) ) )
54anim1d 547 . . 3  |-  ( Rel 
R  ->  ( ( <. x ,  y >.  e.  R  /\  z  =  A )  ->  (
( x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) ) )
65ssoprab2g 25032 . 2  |-  ( Rel 
R  ->  { <. <. x ,  y >. ,  z
>.  |  ( <. x ,  y >.  e.  R  /\  z  =  A
) }  C_  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) } )
7 dmexg 4939 . . 3  |-  ( R  e.  B  ->  dom  R  e.  _V )
8 rnexg 4940 . . 3  |-  ( R  e.  B  ->  ran  R  e.  _V )
9 df-mpt2 5863 . . . . 5  |-  ( x  e.  dom  R , 
y  e.  ran  R  |->  A )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }
109eqcomi 2287 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }  =  ( x  e.  dom  R ,  y  e.  ran  R 
|->  A )
1110mpt2exg 6196 . . 3  |-  ( ( dom  R  e.  _V  /\ 
ran  R  e.  _V )  ->  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }  e.  _V )
127, 8, 11syl2anc 642 . 2  |-  ( R  e.  B  ->  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }  e.  _V )
13 ssexg 4160 . 2  |-  ( ( { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  z  =  A ) }  C_  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }  /\  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  dom  R  /\  y  e.  ran  R )  /\  z  =  A ) }  e.  _V )  ->  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  R  /\  z  =  A ) }  e.  _V )
146, 12, 13syl2anr 464 1  |-  ( ( R  e.  B  /\  Rel  R )  ->  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  R  /\  z  =  A ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694   {coprab 5859    e. cmpt2 5860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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