Theorem ssimaexg 5558

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 4949	. . . . . 6 |- ( y = A -> ( F " y ) = ( F " A ) )
2	1	sseq2d 3177	. . . . 5 |- ( y = A -> ( B C_ ( F " y ) <-> B C_ ( F " A ) ) )
3	2	anbi2d 461	. . . 4 |- ( y = A -> ( ( Fun F /\ B C_ ( F " y ) ) <-> ( Fun F /\ B C_ ( F " A ) ) ) )
4		sseq2 3171	. . . . . 6 |- ( y = A -> ( x C_ y <-> x C_ A ) )
5	4	anbi1d 462	. . . . 5 |- ( y = A -> ( ( x C_ y /\ B = ( F " x ) ) <-> ( x C_ A /\ B = ( F " x ) ) ) )
6	5	exbidv 1818	. . . 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 2733	. . . 4 |- y e. _V
9	8	ssimaex 5557	. . 3 |- ( ( Fun F /\ B C_ ( F " y ) ) -> E. x ( x C_ y /\ B = ( F " x ) ) )
10	7, 9	vtoclg 2790	. 2 |- ( A e. C -> ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) ) )
11	10	3impib 1196	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 973   = wceq 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   "cima 4614   Fun wfun 5192

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tgrest  12963