Theorem releldm2 5831

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 2610	. . 3 |- ( B e. dom A -> B e. _V )
2	1	anim2i 334	. 2 |- ( ( Rel A /\ B e. dom A ) -> ( Rel A /\ B e. _V ) )
3		id 19	. . . . 5 |- ( ( 1st ` x ) = B -> ( 1st ` x ) = B )
4		vex 2604	. . . . . 6 |- x e. _V
5		1stexg 5814	. . . . . 6 |- ( x e. _V -> ( 1st ` x ) e. _V )
6	4, 5	ax-mp 7	. . . . 5 |- ( 1st ` x ) e. _V
7	3, 6	syl6eqelr 2170	. . . 4 |- ( ( 1st ` x ) = B -> B e. _V )
8	7	rexlimivw 2473	. . 3 |- ( E. x e. A ( 1st ` x ) = B -> B e. _V )
9	8	anim2i 334	. 2 |- ( ( Rel A /\ E. x e. A ( 1st ` x ) = B ) -> ( Rel A /\ B e. _V ) )
10		eldm2g 4549	. . . 4 |- ( B e. _V -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
11	10	adantl 271	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
12		df-rel 4370	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
13		ssel 2993	. . . . . . . . 9 |- ( A C_ ( _V X. _V ) -> ( x e. A -> x e. ( _V X. _V ) ) )
14	12, 13	sylbi 119	. . . . . . . 8 |- ( Rel A -> ( x e. A -> x e. ( _V X. _V ) ) )
15	14	imp 122	. . . . . . 7 |- ( ( Rel A /\ x e. A ) -> x e. ( _V X. _V ) )
16		op1steq 5825	. . . . . . 7 |- ( x e. ( _V X. _V ) -> ( ( 1st ` x ) = B <-> E. y x = <. B , y >. ) )
17	15, 16	syl 14	. . . . . 6 |- ( ( Rel A /\ x e. A ) -> ( ( 1st ` x ) = B <-> E. y x = <. B , y >. ) )
18	17	rexbidva 2365	. . . . 5 |- ( Rel A -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
19	18	adantr 270	. . . 4 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
20		rexcom4 2622	. . . . 5 |- ( E. x e. A E. y x = <. B , y >. <-> E. y E. x e. A x = <. B , y >. )
21		risset 2394	. . . . . 6 |- ( <. B , y >. e. A <-> E. x e. A x = <. B , y >. )
22	21	exbii 1536	. . . . 5 |- ( E. y <. B , y >. e. A <-> E. y E. x e. A x = <. B , y >. )
23	20, 22	bitr4i 185	. . . 4 |- ( E. x e. A E. y x = <. B , y >. <-> E. y <. B , y >. e. A )
24	19, 23	syl6bb 194	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. y <. B , y >. e. A ) )
25	11, 24	bitr4d 189	. 2 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )
26	2, 9, 25	pm5.21nd 858	1 |- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   E.wrex 2349   _Vcvv 2601   C_ wss 2973   <.cop 3401   X. cxp 4361   dom cdm 4363   Rel wrel 4368   ` cfv 4922   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by:  reldm  5832