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Theorem elovmpt3imp 6533
 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:   ,

Proof of Theorem elovmpt3imp
StepHypRef Expression
1 ne0i 3790 . 2
2 ax-1 6 . . 3
3 elovmpt3imp.o . . . . 5
43mpt2ndm0 6516 . . . 4
5 fveq1 5870 . . . . 5
6 0fv 5904 . . . . 5
75, 6syl6eq 2514 . . . 4
8 eqneqall 2664 . . . 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 1395  e.wcel 1818  =/=wne 2652   cvv 3109   c0 3784  e.cmpt 4510  cfv 5593  (class class class)co 6296  e.`cmpt2 6298 This theorem is referenced by:  elovmpt3rab1  6536  elovmptnn0wrd  12584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014  df-iota 5556  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301
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