Theorem ssimaexg 5704

Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Assertion

Ref	Expression
ssimaexg	|- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) )

Distinct variable groups:   x, A   x, B   x, F

Allowed substitution hint:   C( x)

Proof of Theorem ssimaexg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq2 5070	. . . . . 6 |- ( y = A -> ( F " y ) = ( F " A ) )
2	1	sseq2d 3255	. . . . 5 |- ( y = A -> ( B C_ ( F " y ) <-> B C_ ( F " A ) ) )
3	2	anbi2d 464	. . . 4 |- ( y = A -> ( ( Fun F /\ B C_ ( F " y ) ) <-> ( Fun F /\ B C_ ( F " A ) ) ) )
4		sseq2 3249	. . . . . 6 |- ( y = A -> ( x C_ y <-> x C_ A ) )
5	4	anbi1d 465	. . . . 5 |- ( y = A -> ( ( x C_ y /\ B = ( F " x ) ) <-> ( x C_ A /\ B = ( F " x ) ) ) )
6	5	exbidv 1871	. . . 4 |- ( y = A -> ( E. x ( x C_ y /\ B = ( F " x ) ) <-> E. x ( x C_ A /\ B = ( F " x ) ) ) )
7	3, 6	imbi12d 234	. . 3 |- ( y = A -> ( ( ( Fun F /\ B C_ ( F " y ) ) -> E. x ( x C_ y /\ B = ( F " x ) ) ) <-> ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) ) ) )
8		vex 2803	. . . 4 |- y e. _V
9	8	ssimaex 5703	. . 3 |- ( ( Fun F /\ B C_ ( F " y ) ) -> E. x ( x C_ y /\ B = ( F " x ) ) )
10	7, 9	vtoclg 2862	. 2 |- ( A e. C -> ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) ) )
11	10	3impib 1225	1 |- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   C_ wss 3198   "cima 4726   Fun wfun 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  tgrest  14883