Theorem 1stvalg 6336

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 4297	. . 3 |- ( A e. _V -> { A } e. _V )
2		dmexg 5021	. . 3 |- ( { A } e. _V -> dom { A } e. _V )
3		uniexg 4560	. . 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 3700	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	dmeqd 4958	. . . 4 |- ( x = A -> dom { x } = dom { A } )
7	6	unieqd 3925	. . 3 |- ( x = A -> U. dom { x } = U. dom { A } )
8		df-1st 6334	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
9	7, 8	fvmptg 5753	. 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 1398   e. wcel 2203   _Vcvv 2813   {csn 3689   U.cuni 3914   dom cdm 4749   ` cfv 5352   1stc1st 6332

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-1st 6334

This theorem is referenced by:  1st0  6338  op1st  6340  elxp6  6363