Theorem mpt2ndm0 6516

Description: The value of an operation given by a maps-to rule is the empty set if the arguments are not contained in the base sets of the rule. (Contributed by Alexander van der Vekens, 12-Oct-2017.)

Hypothesis

Ref	Expression
mpt2ndm0.f

Assertion

Ref	Expression
mpt2ndm0

Distinct variable groups:   ,,   ,,

Proof of Theorem mpt2ndm0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2ndm0.f	. . . . 5
2		df-mpt2 6301	. . . . 5
3	1, 2	eqtri 2486	. . . 4
4	3	dmeqi 5209	. . 3
5		dmoprabss 6384	. . 3
6	4, 5	eqsstri 3533	. 2
7		nssdmovg 6457	. 2
8	6, 7	mpan 670	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  C_wss 3475  c0 3784  X.cxp 5002  domcdm 5004  (class class class)co 6296  {coprab 6297  e.cmpt2 6298

This theorem is referenced by:  elovmpt3imp  6533  bropopvvv  6880  supp0prc  6921  brovex  6969  fullfunc  15275  fthfunc  15276  natfval  15315  evlval  18193  matbas0  18912  matrcl  18914  marrepfval  19062  marepvfval  19067  submafval  19081  minmar1fval  19148  hmeofval  20259  nghmfval  21229  uvtxisvtx  24490  uvtx0  24491  uvtx01vtx  24492  clwwlknprop  24772  2wlkonot3v  24875  2spthonot3v  24876

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