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Theorem elovmpt3imp 6515
Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands must be sets. Remark: a function which is the result of an operation can be regared as operation with 3 operands - therefore the abbreviation "mpt3" is used in the label. (Contributed by AV, 16-May-2019.)
Hypothesis
Ref Expression
elovmpt3imp.o  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  B ) )
Assertion
Ref Expression
elovmpt3imp  |-  ( A  e.  ( ( X O Y ) `  Z )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y, z)    B( x, y, z)    M( x, y, z)    O( x, y, z)    X( x, y, z)    Y( x, y, z)    Z( x, y, z)

Proof of Theorem elovmpt3imp
StepHypRef Expression
1 ne0i 3791 . 2  |-  ( A  e.  ( ( X O Y ) `  Z )  ->  (
( X O Y ) `  Z )  =/=  (/) )
2 ax-1 6 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  ( ( ( X O Y ) `  Z )  =/=  (/)  ->  ( X  e.  _V  /\  Y  e.  _V ) ) )
3 elovmpt3imp.o . . . . 5  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  ( z  e.  M  |->  B ) )
43mpt2ndm0 6498 . . . 4  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  (/) )
5 fveq1 5863 . . . . 5  |-  ( ( X O Y )  =  (/)  ->  ( ( X O Y ) `
 Z )  =  ( (/) `  Z ) )
6 0fv 5897 . . . . 5  |-  ( (/) `  Z )  =  (/)
75, 6syl6eq 2524 . . . 4  |-  ( ( X O Y )  =  (/)  ->  ( ( X O Y ) `
 Z )  =  (/) )
8 eqneqall 2674 . . . 4  |-  ( ( ( X O Y ) `  Z )  =  (/)  ->  ( ( ( X O Y ) `  Z )  =/=  (/)  ->  ( X  e.  _V  /\  Y  e. 
_V ) ) )
94, 7, 83syl 20 . . 3  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( ( ( X O Y ) `  Z )  =/=  (/)  ->  ( X  e.  _V  /\  Y  e.  _V ) ) )
102, 9pm2.61i 164 . 2  |-  ( ( ( X O Y ) `  Z )  =/=  (/)  ->  ( X  e.  _V  /\  Y  e. 
_V ) )
111, 10syl 16 1  |-  ( A  e.  ( ( X O Y ) `  Z )  ->  ( X  e.  _V  /\  Y  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  elovmpt3rab1  6518  elovmptnn0wrd  12545
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