Theorem 2nd0 4074

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 4072	. 2 |- (2nd` (/)) = U.ran {(/)}
2		dmsn0 3319	. . . 4 |- dom {(/)} = (/)
3		dm0rn0 3325	. . . 4 |- (dom {(/)} = (/) <-> ran {(/)} = (/))
4	2, 3	mpbi 189	. . 3 |- ran {(/)} = (/)
5	4	unieqi 2506	. 2 |- U.ran {(/)} = U.(/)
6		uni0 2520	. 2 |- U.(/) = (/)
7	1, 5, 6	3eqtr 1496	1 |- (2nd` (/)) = (/)

Syntax hints:   = wceq 954  (/)c0 2276  {csn 2405  U.cuni 2498  dom cdm 3165  ran crn 3166  ` cfv 3177  2ndc2nd 4068

This theorem is referenced by:  smfval 8176  codval 10536  cmpval 10538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-2nd 4070

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