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

Step	Hyp	Ref	Expression
1		ne0i 3790	. 2
2		ax-1 6	. . 3
3		elovmpt3imp.o	. . . . 5
4	3	mpt2ndm0 6516	. . . 4
5		fveq1 5870	. . . . 5
6		0fv 5904	. . . . 5
7	5, 6	syl6eq 2514	. . . 4
8		eqneqall 2664	. . . 4
9	4, 7, 8	3syl 20	. . 3
10	2, 9	pm2.61i 164	. 2
11	1, 10	syl 16	1

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