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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
Assertion
Ref Expression
elovmpt3imp
Distinct variable group:   ,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)   (,,)   (,,)   (,,)

Proof of Theorem elovmpt3imp
StepHypRef Expression
1 ne0i 3774 . 2
2 ax-1 6 . . 3
3 elovmpt3imp.o . . . . 5
43mpt2ndm0 6498 . . . 4
5 fveq1 5852 . . . . 5
6 0fv 5886 . . . . 5
75, 6syl6eq 2498 . . . 4
8 eqneqall 2648 . . . 4
94, 7, 83syl 20 . . 3
102, 9pm2.61i 164 . 2
111, 10syl 16 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   wceq 1381   wcel 1802   wne 2636  cvv 3093  c0 3768   cmpt 4492  cfv 5575  (class class class)co 6278   cmpt2 6280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-xp 4992  df-dm 4996  df-iota 5538  df-fv 5583  df-ov 6281  df-oprab 6282  df-mpt2 6283 This theorem is referenced by:  elovmpt3rab1  6518  elovmptnn0wrd  12560
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