Theorem releldm2 5811

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 2566	. . 3 |- ( B e. dom A -> B e. _V )
2	1	anim2i 324	. 2 |- ( ( Rel A /\ B e. dom A ) -> ( Rel A /\ B e. _V ) )
3		id 19	. . . . 5 |- ( ( 1st ` x ) = B -> ( 1st ` x ) = B )
4		vex 2560	. . . . . 6 |- x e. _V
5		1stexg 5794	. . . . . 6 |- ( x e. _V -> ( 1st ` x ) e. _V )
6	4, 5	ax-mp 7	. . . . 5 |- ( 1st ` x ) e. _V
7	3, 6	syl6eqelr 2129	. . . 4 |- ( ( 1st ` x ) = B -> B e. _V )
8	7	rexlimivw 2429	. . 3 |- ( E. x e. A ( 1st ` x ) = B -> B e. _V )
9	8	anim2i 324	. 2 |- ( ( Rel A /\ E. x e. A ( 1st ` x ) = B ) -> ( Rel A /\ B e. _V ) )
10		eldm2g 4531	. . . 4 |- ( B e. _V -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
11	10	adantl 262	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
12		df-rel 4352	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
13		ssel 2939	. . . . . . . . 9 |- ( A C_ ( _V X. _V ) -> ( x e. A -> x e. ( _V X. _V ) ) )
14	12, 13	sylbi 114	. . . . . . . 8 |- ( Rel A -> ( x e. A -> x e. ( _V X. _V ) ) )
15	14	imp 115	. . . . . . 7 |- ( ( Rel A /\ x e. A ) -> x e. ( _V X. _V ) )
16		op1steq 5805	. . . . . . 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 2323	. . . . 5 |- ( Rel A -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
19	18	adantr 261	. . . 4 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
20		rexcom4 2577	. . . . 5 |- ( E. x e. A E. y x = <. B , y >. <-> E. y E. x e. A x = <. B , y >. )
21		risset 2352	. . . . . 6 |- ( <. B , y >. e. A <-> E. x e. A x = <. B , y >. )
22	21	exbii 1496	. . . . 5 |- ( E. y <. B , y >. e. A <-> E. y E. x e. A x = <. B , y >. )
23	20, 22	bitr4i 176	. . . 4 |- ( E. x e. A E. y x = <. B , y >. <-> E. y <. B , y >. e. A )
24	19, 23	syl6bb 185	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. y <. B , y >. e. A ) )
25	11, 24	bitr4d 180	. 2 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )
26	2, 9, 25	pm5.21nd 825	1 |- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   _Vcvv 2557   C_ wss 2917   <.cop 3378   X. cxp 4343   dom cdm 4345   Rel wrel 4350   ` cfv 4902   1stc1st 5765

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by:  reldm  5812