Theorem relelfvdm 5205

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 5176	. . . . . 6 |- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )
2		exsimpr 1509	. . . . . 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 114	. . . . 5 |- ( A e. ( F ` B ) -> E. x A. y ( B F y <-> y = x ) )
4		equsb1 1668	. . . . . . . 8 |- [ x / y ] y = x
5		spsbbi 1725	. . . . . . . 8 |- ( A. y ( B F y <-> y = x ) -> ( [ x / y ] B F y <-> [ x / y ] y = x ) )
6	4, 5	mpbiri 157	. . . . . . 7 |- ( A. y ( B F y <-> y = x ) -> [ x / y ] B F y )
7		nfv 1421	. . . . . . . 8 |- F/ y B F x
8		breq2 3768	. . . . . . . 8 |- ( y = x -> ( B F y <-> B F x ) )
9	7, 8	sbie 1674	. . . . . . 7 |- ( [ x / y ] B F y <-> B F x )
10	6, 9	sylib 127	. . . . . 6 |- ( A. y ( B F y <-> y = x ) -> B F x )
11	10	eximi 1491	. . . . 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 324	. . 3 |- ( ( Rel F /\ A e. ( F ` B ) ) -> ( Rel F /\ E. x B F x ) )
14		19.42v 1786	. . 3 |- ( E. x ( Rel F /\ B F x ) <-> ( Rel F /\ E. x B F x ) )
15	13, 14	sylibr 137	. 2 |- ( ( Rel F /\ A e. ( F ` B ) ) -> E. x ( Rel F /\ B F x ) )
16		releldm 4569	. . 3 |- ( ( Rel F /\ B F x ) -> B e. dom F )
17	16	exlimiv 1489	. 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 97   <-> wb 98   A.wal 1241   E.wex 1381   e. wcel 1393   [wsb 1645   class class class wbr 3764   dom cdm 4345   Rel wrel 4350   ` cfv 4902

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-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

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  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-xp 4351  df-rel 4352  df-dm 4355  df-iota 4867  df-fv 4910

This theorem is referenced by:  elmpt2cl  5698  mpt2xopn0yelv  5854  eluzel2  8478