Theorem ssimaexg 5664

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 5037	. . . . . 6 |- ( y = A -> ( F " y ) = ( F " A ) )
2	1	sseq2d 3231	. . . . 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 3225	. . . . . 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 1849	. . . 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 2779	. . . 4 |- y e. _V
9	8	ssimaex 5663	. . 3 |- ( ( Fun F /\ B C_ ( F " y ) ) -> E. x ( x C_ y /\ B = ( F " x ) ) )
10	7, 9	vtoclg 2838	. 2 |- ( A e. C -> ( ( Fun F /\ B C_ ( F " A ) ) -> E. x ( x C_ A /\ B = ( F " x ) ) ) )
11	10	3impib 1204	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 981   = wceq 1373   E.wex 1516   e. wcel 2178   C_ wss 3174   "cima 4696   Fun wfun 5284

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  tgrest  14756