Theorem 1st2val 7194

Description: Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)

Assertion

Ref	Expression
1st2val	|- ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A )

Distinct variable group:   x, y, z

Allowed substitution hints:   A( x, y, z)

Proof of Theorem 1st2val

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5177	. . 3 |- ( A e. ( _V X. _V ) <-> E. w E. v A = <. w , v >. )
2		fveq2 6191	. . . . . 6 |- ( A = <. w , v >. -> ( { <. <. x , y >. , z >. | z = x } ` A ) = ( { <. <. x , y >. , z >. | z = x } ` <. w , v >. ) )
3		df-ov 6653	. . . . . . 7 |- ( w { <. <. x , y >. , z >. | z = x } v ) = ( { <. <. x , y >. , z >. | z = x } ` <. w , v >. )
4		vex 3203	. . . . . . . 8 |- w e. _V
5		vex 3203	. . . . . . . 8 |- v e. _V
6		simpl 473	. . . . . . . . 9 |- ( ( x = w /\ y = v ) -> x = w )
7		mpt2v 6750	. . . . . . . . . 10 |- ( x e. _V , y e. _V |-> x ) = { <. <. x , y >. , z >. | z = x }
8	7	eqcomi 2631	. . . . . . . . 9 |- { <. <. x , y >. , z >. | z = x } = ( x e. _V , y e. _V |-> x )
9	6, 8, 4	ovmpt2a 6791	. . . . . . . 8 |- ( ( w e. _V /\ v e. _V ) -> ( w { <. <. x , y >. , z >. | z = x } v ) = w )
10	4, 5, 9	mp2an 708	. . . . . . 7 |- ( w { <. <. x , y >. , z >. | z = x } v ) = w
11	3, 10	eqtr3i 2646	. . . . . 6 |- ( { <. <. x , y >. , z >. | z = x } ` <. w , v >. ) = w
12	2, 11	syl6eq 2672	. . . . 5 |- ( A = <. w , v >. -> ( { <. <. x , y >. , z >. | z = x } ` A ) = w )
13	4, 5	op1std 7178	. . . . 5 |- ( A = <. w , v >. -> ( 1st ` A ) = w )
14	12, 13	eqtr4d 2659	. . . 4 |- ( A = <. w , v >. -> ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A ) )
15	14	exlimivv 1860	. . 3 |- ( E. w E. v A = <. w , v >. -> ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A ) )
16	1, 15	sylbi 207	. 2 |- ( A e. ( _V X. _V ) -> ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A ) )
17		vex 3203	. . . . . . . . . 10 |- x e. _V
18		vex 3203	. . . . . . . . . 10 |- y e. _V
19	17, 18	pm3.2i 471	. . . . . . . . 9 |- ( x e. _V /\ y e. _V )
20		ax6ev 1890	. . . . . . . . 9 |- E. z z = x
21	19, 20	2th 254	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V ) <-> E. z z = x )
22	21	opabbii 4717	. . . . . . 7 |- { <. x , y >. | ( x e. _V /\ y e. _V ) } = { <. x , y >. | E. z z = x }
23		df-xp 5120	. . . . . . 7 |- ( _V X. _V ) = { <. x , y >. | ( x e. _V /\ y e. _V ) }
24		dmoprab 6741	. . . . . . 7 |- dom { <. <. x , y >. , z >. | z = x } = { <. x , y >. | E. z z = x }
25	22, 23, 24	3eqtr4ri 2655	. . . . . 6 |- dom { <. <. x , y >. , z >. | z = x } = ( _V X. _V )
26	25	eleq2i 2693	. . . . 5 |- ( A e. dom { <. <. x , y >. , z >. | z = x } <-> A e. ( _V X. _V ) )
27		ndmfv 6218	. . . . 5 |- ( -. A e. dom { <. <. x , y >. , z >. | z = x } -> ( { <. <. x , y >. , z >. | z = x } ` A ) = (/) )
28	26, 27	sylnbir 321	. . . 4 |- ( -. A e. ( _V X. _V ) -> ( { <. <. x , y >. , z >. | z = x } ` A ) = (/) )
29		dmsnn0 5600	. . . . . . . 8 |- ( A e. ( _V X. _V ) <-> dom { A } =/= (/) )
30	29	biimpri 218	. . . . . . 7 |- ( dom { A } =/= (/) -> A e. ( _V X. _V ) )
31	30	necon1bi 2822	. . . . . 6 |- ( -. A e. ( _V X. _V ) -> dom { A } = (/) )
32	31	unieqd 4446	. . . . 5 |- ( -. A e. ( _V X. _V ) -> U. dom { A } = U. (/) )
33		uni0 4465	. . . . 5 |- U. (/) = (/)
34	32, 33	syl6eq 2672	. . . 4 |- ( -. A e. ( _V X. _V ) -> U. dom { A } = (/) )
35	28, 34	eqtr4d 2659	. . 3 |- ( -. A e. ( _V X. _V ) -> ( { <. <. x , y >. , z >. | z = x } ` A ) = U. dom { A } )
36		1stval 7170	. . 3 |- ( 1st ` A ) = U. dom { A }
37	35, 36	syl6eqr 2674	. 2 |- ( -. A e. ( _V X. _V ) -> ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A ) )
38	16, 37	pm2.61i 176	1 |- ( { <. <. x , y >. , z >. | z = x } ` A ) = ( 1st ` A )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   {copab 4712   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   {coprab 6651   |-> cmpt2 6652   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168

This theorem is referenced by: (None)