Theorem 1stvalg 6121

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 4170	. . 3 |- ( A e. _V -> { A } e. _V )
2		dmexg 4875	. . 3 |- ( { A } e. _V -> dom { A } e. _V )
3		uniexg 4424	. . 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 3594	. . . . 5 |- ( x = A -> { x } = { A } )
6	5	dmeqd 4813	. . . 4 |- ( x = A -> dom { x } = dom { A } )
7	6	unieqd 3807	. . 3 |- ( x = A -> U. dom { x } = U. dom { A } )
8		df-1st 6119	. . 3 |- 1st = ( x e. _V |-> U. dom { x } )
9	7, 8	fvmptg 5572	. 2 |- ( ( A e. _V /\ U. dom { A } e. _V ) -> ( 1st ` A ) = U. dom { A } )
10	4, 9	mpdan 419	1 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   {csn 3583   U.cuni 3796   dom cdm 4611   ` cfv 5198   1stc1st 6117

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119

This theorem is referenced by:  1st0  6123  op1st  6125  elxp6  6148