Theorem ssimaexg 5476

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 4872	. . . . . 6 |- ( y = A -> ( F " y ) = ( F " A ) )
2	1	sseq2d 3122	. . . . 5 |- ( y = A -> ( B C_ ( F " y ) <-> B C_ ( F " A ) ) )
3	2	anbi2d 459	. . . 4 |- ( y = A -> ( ( Fun F /\ B C_ ( F " y ) ) <-> ( Fun F /\ B C_ ( F " A ) ) ) )
4		sseq2 3116	. . . . . 6 |- ( y = A -> ( x C_ y <-> x C_ A ) )
5	4	anbi1d 460	. . . . 5 |- ( y = A -> ( ( x C_ y /\ B = ( F " x ) ) <-> ( x C_ A /\ B = ( F " x ) ) ) )
6	5	exbidv 1797	. . . 4 |- ( y = A -> ( E. x ( x C_ y /\ B = ( F " x ) ) <-> E. x ( x C_ A /\ B = ( F " x ) ) ) )
7	3, 6	imbi12d 233	. . 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 2684	. . . 4 |- y e. _V
9	8	ssimaex 5475	. . 3 |- ( ( Fun F /\ B C_ ( F " y ) ) -> E. x ( x C_ y /\ B = ( F " x ) ) )
10	7, 9	vtoclg 2741	. 2 |- ( A e. C -> ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) ) )
11	10	3impib 1179	1 |- ( ( A e. C /\ Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   C_ wss 3066   "cima 4537   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  tgrest  12327