Theorem 2nd0 5025

Description: The value of the second-member function at the empty set.

Assertion

Ref	Expression
2nd0	|- (2nd` (/)) = (/)

Proof of Theorem 2nd0

Step	Hyp	Ref	Expression
1		2ndval 5023	. 2 |- (2nd` (/)) = U.ran {(/)}
2		dmsn0 4365	. . . 4 |- dom {(/)} = (/)
3		dm0rn0 4175	. . . 4 |- (dom {(/)} = (/) <-> ran {(/)} = (/))
4	2, 3	mpbi 206	. . 3 |- ran {(/)} = (/)
5	4	unieqi 3187	. 2 |- U.ran {(/)} = U.(/)
6		uni0 3205	. 2 |- U.(/) = (/)
7	1, 5, 6	3eqtri 1912	1 |- (2nd` (/)) = (/)

Syntax hints:   = wceq 1298  (/)c0 2875  {csn 3044  U.cuni 3177  dom cdm 3986  ran crn 3987  ` cfv 3998  2ndc2nd 5019

This theorem is referenced by:  smfval 9556  codval 15071  cmpval 15073

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-2nd 5021

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