Theorem 1stvalg 6307

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 4273	. . 3 |- ( A e. _V -> { A } e. _V )
2		dmexg 4995	. . 3 |- ( { A } e. _V -> dom { A } e. _V )
3		uniexg 4535	. . 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 3679	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	dmeqd 4932	. . . 4 |- ( x = A -> dom { x } = dom { A } )
7	6	unieqd 3903	. . 3 |- ( x = A -> U. dom { x } = U. dom { A } )
8		df-1st 6305	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
9	7, 8	fvmptg 5722	. 2 |- ( ( A e. _V /\ U. dom { A } e. _V ) -> ( 1st ` A ) = U. dom { A } )
10	4, 9	mpdan 421	1 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   _Vcvv 2801   {csn 3668   U.cuni 3892   dom cdm 4724   ` cfv 5325   1stc1st 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-iota 5285  df-fun 5327  df-fv 5333  df-1st 6305

This theorem is referenced by:  1st0  6309  op1st  6311  elxp6  6334