Theorem 1stvalg 6048

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 4116	. . 3 |- ( A e. _V -> { A } e. _V )
2		dmexg 4811	. . 3 |- ( { A } e. _V -> dom { A } e. _V )
3		uniexg 4369	. . 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 3543	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	dmeqd 4749	. . . 4 |- ( x = A -> dom { x } = dom { A } )
7	6	unieqd 3755	. . 3 |- ( x = A -> U. dom { x } = U. dom { A } )
8		df-1st 6046	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
9	7, 8	fvmptg 5505	. 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 1332   e. wcel 1481   _Vcvv 2689   {csn 3532   U.cuni 3744   dom cdm 4547   ` cfv 5131   1stc1st 6044

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-1st 6046

This theorem is referenced by:  1st0  6050  op1st  6052  elxp6  6075