Theorem releldm2 6392

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 2827	. . 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 2818	. . . . . 6 |- x e. _V
5		1stexg 6374	. . . . . 6 |- ( x e. _V -> ( 1st ` x ) e. _V )
6	4, 5	ax-mp 5	. . . . 5 |- ( 1st ` x ) e. _V
7	3, 6	eqeltrrdi 2326	. . . 4 |- ( ( 1st ` x ) = B -> B e. _V )
8	7	rexlimivw 2658	. . 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 4957	. . . 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 4761	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
13		ssel 3236	. . . . . . . . 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 6386	. . . . . . 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 2541	. . . . 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 2839	. . . . 5 |- ( E. x e. A E. y x = <. B , y >. <-> E. y E. x e. A x = <. B , y >. )
21		risset 2572	. . . . . 6 |- ( <. B , y >. e. A <-> E. x e. A x = <. B , y >. )
22	21	exbii 1654	. . . . 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 924	1 |- ( Rel A -> ( B e. dom A <-> E. x e. A ( 1st ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   _Vcvv 2815   C_ wss 3214   <.cop 3697   X. cxp 4752   dom cdm 4754   Rel wrel 4759   ` cfv 5357   1stc1st 6345

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-1st 6347  df-2nd 6348

This theorem is referenced by:  reldm  6393