Theorem relelfvdm 5226

Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)

Assertion

Ref	Expression
relelfvdm	|- ( ( Rel F /\ A e. ( F ` B ) ) -> B e. dom F )

Proof of Theorem relelfvdm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfv 5196	. . . . . 6 |- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )
2		exsimpr 1549	. . . . . 6 |- ( E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) -> E. x A. y ( B F y <-> y = x ) )
3	1, 2	sylbi 119	. . . . 5 |- ( A e. ( F ` B ) -> E. x A. y ( B F y <-> y = x ) )
4		equsb1 1708	. . . . . . . 8 |- [ x / y ] y = x
5		spsbbi 1765	. . . . . . . 8 |- ( A. y ( B F y <-> y = x ) -> ( [ x / y ] B F y <-> [ x / y ] y = x ) )
6	4, 5	mpbiri 166	. . . . . . 7 |- ( A. y ( B F y <-> y = x ) -> [ x / y ] B F y )
7		nfv 1461	. . . . . . . 8 |- F/ y B F x
8		breq2 3789	. . . . . . . 8 |- ( y = x -> ( B F y <-> B F x ) )
9	7, 8	sbie 1714	. . . . . . 7 |- ( [ x / y ] B F y <-> B F x )
10	6, 9	sylib 120	. . . . . 6 |- ( A. y ( B F y <-> y = x ) -> B F x )
11	10	eximi 1531	. . . . 5 |- ( E. x A. y ( B F y <-> y = x ) -> E. x B F x )
12	3, 11	syl 14	. . . 4 |- ( A e. ( F ` B ) -> E. x B F x )
13	12	anim2i 334	. . 3 |- ( ( Rel F /\ A e. ( F ` B ) ) -> ( Rel F /\ E. x B F x ) )
14		19.42v 1827	. . 3 |- ( E. x ( Rel F /\ B F x ) <-> ( Rel F /\ E. x B F x ) )
15	13, 14	sylibr 132	. 2 |- ( ( Rel F /\ A e. ( F ` B ) ) -> E. x ( Rel F /\ B F x ) )
16		releldm 4587	. . 3 |- ( ( Rel F /\ B F x ) -> B e. dom F )
17	16	exlimiv 1529	. 2 |- ( E. x ( Rel F /\ B F x ) -> B e. dom F )
18	15, 17	syl 14	1 |- ( ( Rel F /\ A e. ( F ` B ) ) -> B e. dom F )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   e. wcel 1433   [wsb 1685   class class class wbr 3785   dom cdm 4363   Rel wrel 4368   ` cfv 4922

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-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

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  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-xp 4369  df-rel 4370  df-dm 4373  df-iota 4887  df-fv 4930

This theorem is referenced by:  elmpt2cl  5718  mpt2xopn0yelv  5877  eluzel2  8624