Theorem releldm2 5753

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 2560	. . 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 2554	. . . . . 6 |- x e. _V
5		1stexg 5736	. . . . . 6 |- (x e. _V -> ( 1st ` x) e. _V)
6	4, 5	ax-mp 7	. . . . 5 |- ( 1st ` x) e. _V
7	3, 6	syl6eqelr 2126	. . . 4 |- (( 1st ` x) = B -> B e. _V)
8	7	rexlimivw 2423	. . 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 4474	. . . 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 4295	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V))
13		ssel 2933	. . . . . . . . 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 5747	. . . . . . 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 2317	. . . . 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 2571	. . . . 5 |- (E.x e. A E.y x = <.B , y >. <-> E.yE.x e. A x = <.B , y >.)
21		risset 2346	. . . . . 6 |- ( <.B , y >. e. A <-> E.x e. A x = <.B , y >.)
22	21	exbii 1493	. . . . 5 |- (E.y <.B , y >. e. A <-> E.yE.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 824	1 |- ( Rel A -> (B e. dom A <-> E.x e. A ( 1st ` x) = B))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  E.wrex 2301   _Vcvv 2551   C_ wss 2911   <.cop 3370   X. cxp 4286   dom cdm 4288   Rel wrel 4293   ` cfv 4845   1stc1st 5707

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710

This theorem is referenced by:  reldm  5754