Theorem 1st0 7174

Description: The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Assertion

Ref	Expression
1st0	|- ( 1st ` (/) ) = (/)

Proof of Theorem 1st0

Step	Hyp	Ref	Expression
1		1stval 7170	. 2 |- ( 1st ` (/) ) = U. dom { (/) }
2		dmsn0 5602	. . 3 |- dom { (/) } = (/)
3	2	unieqi 4445	. 2 |- U. dom { (/) } = U. (/)
4		uni0 4465	. 2 |- U. (/) = (/)
5	1, 3, 4	3eqtri 2648	1 |- ( 1st ` (/) ) = (/)

Syntax hints:   = wceq 1483   (/)c0 3915   {csn 4177   U.cuni 4436   dom cdm 5114   ` cfv 5888   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168

This theorem is referenced by:  vafval  27458