Theorem releldm2 6294

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 2788	. . 3 |- ( B e. dom A -> B e. _V )
2	1	anim2i 342	. 2 |- ( ( Rel A /\ B e. dom A ) -> ( Rel A /\ B e. _V ) )
3		id 19	. . . . 5 |- ( ( 1st ` x ) = B -> ( 1st ` x ) = B )
4		vex 2779	. . . . . 6 |- x e. _V
5		1stexg 6276	. . . . . 6 |- ( x e. _V -> ( 1st ` x ) e. _V )
6	4, 5	ax-mp 5	. . . . 5 |- ( 1st ` x ) e. _V
7	3, 6	eqeltrrdi 2299	. . . 4 |- ( ( 1st ` x ) = B -> B e. _V )
8	7	rexlimivw 2621	. . 3 |- ( E. x e. A ( 1st ` x ) = B -> B e. _V )
9	8	anim2i 342	. 2 |- ( ( Rel A /\ E. x e. A ( 1st ` x ) = B ) -> ( Rel A /\ B e. _V ) )
10		eldm2g 4893	. . . 4 |- ( B e. _V -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
11	10	adantl 277	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. y <. B , y >. e. A ) )
12		df-rel 4700	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
13		ssel 3195	. . . . . . . . 9 |- ( A C_ ( _V X. _V ) -> ( x e. A -> x e. ( _V X. _V ) ) )
14	12, 13	sylbi 121	. . . . . . . 8 |- ( Rel A -> ( x e. A -> x e. ( _V X. _V ) ) )
15	14	imp 124	. . . . . . 7 |- ( ( Rel A /\ x e. A ) -> x e. ( _V X. _V ) )
16		op1steq 6288	. . . . . . 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 2505	. . . . 5 |- ( Rel A -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
19	18	adantr 276	. . . 4 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. x e. A E. y x = <. B , y >. ) )
20		rexcom4 2800	. . . . 5 |- ( E. x e. A E. y x = <. B , y >. <-> E. y E. x e. A x = <. B , y >. )
21		risset 2536	. . . . . 6 |- ( <. B , y >. e. A <-> E. x e. A x = <. B , y >. )
22	21	exbii 1629	. . . . 5 |- ( E. y <. B , y >. e. A <-> E. y E. x e. A x = <. B , y >. )
23	20, 22	bitr4i 187	. . . 4 |- ( E. x e. A E. y x = <. B , y >. <-> E. y <. B , y >. e. A )
24	19, 23	bitrdi 196	. . 3 |- ( ( Rel A /\ B e. _V ) -> ( E. x e. A ( 1st ` x ) = B <-> E. y <. B , y >. e. A ) )
25	11, 24	bitr4d 191	. 2 |- ( ( Rel A /\ B e. _V ) -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )
26	2, 9, 25	pm5.21nd 918	1 |- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   E.wrex 2487   _Vcvv 2776   C_ wss 3174   <.cop 3646   X. cxp 4691   dom cdm 4693   Rel wrel 4698   ` cfv 5290   1stc1st 6247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-1st 6249  df-2nd 6250

This theorem is referenced by:  reldm  6295