Theorem releldm2 7218

Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Assertion

Ref	Expression
releldm2	|- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem releldm2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . 3 |- ( B e. dom A -> B e. _V )
2	1	anim2i 593	. 2 |- ( ( Rel A /\ B e. dom A ) -> ( Rel A /\ B e. _V ) )
3		id 22	. . . . 5 |- ( ( 1st ` x ) = B -> ( 1st ` x ) = B )
4		fvex 6201	. . . . 5 |- ( 1st ` x ) e. _V
5	3, 4	syl6eqelr 2710	. . . 4 |- ( ( 1st ` x ) = B -> B e. _V )
6	5	rexlimivw 3029	. . 3 |- ( E. x e. A ( 1st ` x ) = B -> B e. _V )
7	6	anim2i 593	. 2 |- ( ( Rel A /\ E. x e. A ( 1st ` x ) = B ) -> ( Rel A /\ B e. _V ) )
8		eldm2g 5320	. . . 4 |- ( B e. _V -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
9	8	adantl 482	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
10		df-rel 5121	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
11		ssel 3597	. . . . . . . . 9 |- ( A C_ ( _V X. _V ) -> ( x e. A -> x e. ( _V X. _V ) ) )
12	10, 11	sylbi 207	. . . . . . . 8 |- ( Rel A -> ( x e. A -> x e. ( _V X. _V ) ) )
13	12	imp 445	. . . . . . 7 |- ( ( Rel A /\ x e. A ) -> x e. ( _V X. _V ) )
14		op1steq 7210	. . . . . . 7 |- ( x e. ( _V X. _V ) -> ( ( 1st ` x ) = B <-> E. y x = <. B , y >. ) )
15	13, 14	syl 17	. . . . . 6 |- ( ( Rel A /\ x e. A ) -> ( ( 1st ` x ) = B <-> E. y x = <. B , y >. ) )
16	15	rexbidva 3049	. . . . 5 |- ( Rel A -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
17	16	adantr 481	. . . 4 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
18		rexcom4 3225	. . . . 5 |- ( E. x e. A E. y x = <. B , y >. <-> E. y E. x e. A x = <. B , y >. )
19		risset 3062	. . . . . 6 |- ( <. B , y >. e. A <-> E. x e. A x = <. B , y >. )
20	19	exbii 1774	. . . . 5 |- ( E. y <. B , y >. e. A <-> E. y E. x e. A x = <. B , y >. )
21	18, 20	bitr4i 267	. . . 4 |- ( E. x e. A E. y x = <. B , y >. <-> E. y <. B , y >. e. A )
22	17, 21	syl6bb 276	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. y <. B , y >. e. A ) )
23	9, 22	bitr4d 271	. 2 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )
24	2, 7, 23	pm5.21nd 941	1 |- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   <.cop 4183   X. cxp 5112   dom cdm 5114   Rel wrel 5119   ` cfv 5888   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-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-1st 7168  df-2nd 7169

This theorem is referenced by:  reldm  7219