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Theorem oprabexdOLD 25897
Description: Existence of an operator abstraction. (Moved to oprabexd 6086 in main set.mm and may be deleted by mathbox owner, JM. --NM 16-Sep-2013.) (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
oprabexdOLD.1  |-  ( ph  ->  A  e.  _V )
oprabexdOLD.2  |-  ( ph  ->  B  e.  _V )
oprabexdOLD.3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )
oprabexdOLD.4  |-  ( ph  ->  F  =  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ps ) } )
Assertion
Ref Expression
oprabexdOLD  |-  ( ph  ->  F  e.  _V )
Distinct variable groups:    x, A, y, z    x, B, y, z    ph, x, y, z
Allowed substitution hints:    ps( x, y, z)    F( x, y, z)

Proof of Theorem oprabexdOLD
StepHypRef Expression
1 oprabexdOLD.1 . 2  |-  ( ph  ->  A  e.  _V )
2 oprabexdOLD.2 . 2  |-  ( ph  ->  B  e.  _V )
3 oprabexdOLD.3 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  E* z ps )
4 oprabexdOLD.4 . 2  |-  ( ph  ->  F  =  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ps ) } )
51, 2, 3, 4oprabexd 6086 1  |-  ( ph  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   E*wmo 2218   _Vcvv 2873   {coprab 5982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-oprab 5985
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