Theorem 1stvalg 6084

Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
1stvalg	|- ( A e. _V -> ( 1st ` A ) = U. dom { A } )

Proof of Theorem 1stvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snexg 4144	. . 3 |- ( A e. _V -> { A } e. _V )
2		dmexg 4847	. . 3 |- ( { A } e. _V -> dom { A } e. _V )
3		uniexg 4398	. . 3 |- ( dom { A } e. _V -> U. dom { A } e. _V )
4	1, 2, 3	3syl 17	. 2 |- ( A e. _V -> U. dom { A } e. _V )
5		sneq 3571	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	dmeqd 4785	. . . 4 |- ( x = A -> dom { x } = dom { A } )
7	6	unieqd 3783	. . 3 |- ( x = A -> U. dom { x } = U. dom { A } )
8		df-1st 6082	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
9	7, 8	fvmptg 5541	. 2 |- ( ( A e. _V /\ U. dom { A } e. _V ) -> ( 1st ` A ) = U. dom { A } )
10	4, 9	mpdan 418	1 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   _Vcvv 2712   {csn 3560   U.cuni 3772   dom cdm 4583   ` cfv 5167   1stc1st 6080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-1st 6082

This theorem is referenced by:  1st0  6086  op1st  6088  elxp6  6111