Theorem f1stres 7190

Description: Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
f1stres	|- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A

Proof of Theorem f1stres

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- y e. _V
2		vex 3203	. . . . . . . 8 |- z e. _V
3	1, 2	op1sta 5617	. . . . . . 7 |- U. dom { <. y , z >. } = y
4	3	eleq1i 2692	. . . . . 6 |- ( U. dom { <. y , z >. } e. A <-> y e. A )
5	4	biimpri 218	. . . . 5 |- ( y e. A -> U. dom { <. y , z >. } e. A )
6	5	adantr 481	. . . 4 |- ( ( y e. A /\ z e. B ) -> U. dom { <. y , z >. } e. A )
7	6	rgen2 2975	. . 3 |- A. y e. A A. z e. B U. dom { <. y , z >. } e. A
8		sneq 4187	. . . . . . 7 |- ( x = <. y , z >. -> { x } = { <. y , z >. } )
9	8	dmeqd 5326	. . . . . 6 |- ( x = <. y , z >. -> dom { x } = dom { <. y , z >. } )
10	9	unieqd 4446	. . . . 5 |- ( x = <. y , z >. -> U. dom { x } = U. dom { <. y , z >. } )
11	10	eleq1d 2686	. . . 4 |- ( x = <. y , z >. -> ( U. dom { x } e. A <-> U. dom { <. y , z >. } e. A ) )
12	11	ralxp 5263	. . 3 |- ( A. x e. ( A X. B ) U. dom { x } e. A <-> A. y e. A A. z e. B U. dom { <. y , z >. } e. A )
13	7, 12	mpbir 221	. 2 |- A. x e. ( A X. B ) U. dom { x } e. A
14		df-1st 7168	. . . . 5 |- 1st = ( x e. _V |-> U. dom { x } )
15	14	reseq1i 5392	. . . 4 |- ( 1st |` ( A X. B ) ) = ( ( x e. _V |-> U. dom { x } ) |` ( A X. B ) )
16		ssv 3625	. . . . 5 |- ( A X. B ) C_ _V
17		resmpt 5449	. . . . 5 |- ( ( A X. B ) C_ _V -> ( ( x e. _V |-> U. dom { x } ) |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. dom { x } ) )
18	16, 17	ax-mp 5	. . . 4 |- ( ( x e. _V |-> U. dom { x } ) |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. dom { x } )
19	15, 18	eqtri 2644	. . 3 |- ( 1st |` ( A X. B ) ) = ( x e. ( A X. B ) |-> U. dom { x } )
20	19	fmpt 6381	. 2 |- ( A. x e. ( A X. B ) U. dom { x } e. A <-> ( 1st |` ( A X. B ) ) : ( A X. B ) --> A )
21	13, 20	mpbi 220	1 |- ( 1st |` ( A X. B ) ) : ( A X. B ) --> A

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   |` cres 5116   -->wf 5884   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168

This theorem is referenced by:  fo1stres  7192  1stcof  7196  fparlem1  7277  domssex2  8120  domssex  8121  unxpwdom2  8493  1stfcl  16837  tx1cn  21412  xpinpreima  29952  xpinpreima2  29953  1stmbfm  30322  hausgraph  37790