Theorem ssimaex 5177

Description: The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Hypothesis

Ref	Expression
ssimaex.1	|- A e. _V

Assertion

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

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

Proof of Theorem ssimaex

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmres 4575	. . . . 5 |- dom ( F |` A) = (A i^i dom F)
2	1	imaeq2i 4609	. . . 4 |- ( F " dom ( F |` A)) = ( F "(A i^i dom F))
3		imadmres 4756	. . . 4 |- ( F " dom ( F |` A)) = ( F "A)
4	2, 3	eqtr3i 2059	. . 3 |- ( F "(A i^i dom F)) = ( F "A)
5	4	sseq2i 2964	. 2 |- (B C_ ( F "(A i^i dom F)) <-> B C_ ( F "A))
6		ssrab2 3019	. . . 4 |- {y e. (A i^i dom F) | ( F ` y) e. B } C_ (A i^i dom F)
7		ssel2 2934	. . . . . . . . 9 |- ((B C_ ( F "(A i^i dom F)) /\ z e. B) -> z e. ( F "(A i^i dom F)))
8	7	adantll 445	. . . . . . . 8 |- ((( Fun F /\ B C_ ( F "(A i^i dom F))) /\ z e. B) -> z e. ( F "(A i^i dom F)))
9		fvelima 5168	. . . . . . . . . . . 12 |- (( Fun F /\ z e. ( F "(A i^i dom F))) -> E.w e. (A i^i dom F)( F ` w) = z)
10	9	ex 108	. . . . . . . . . . 11 |- ( Fun F -> (z e. ( F "(A i^i dom F)) -> E.w e. (A i^i dom F)( F ` w) = z))
11	10	adantr 261	. . . . . . . . . 10 |- (( Fun F /\ z e. B) -> (z e. ( F "(A i^i dom F)) -> E.w e. (A i^i dom F)( F ` w) = z))
12		eleq1a 2106	. . . . . . . . . . . . . . . 16 |- (z e. B -> (( F ` w) = z -> ( F ` w) e. B))
13	12	anim2d 320	. . . . . . . . . . . . . . 15 |- (z e. B -> ((w e. (A i^i dom F) /\ ( F ` w) = z) -> (w e. (A i^i dom F) /\ ( F ` w) e. B)))
14		fveq2 5121	. . . . . . . . . . . . . . . . 17 |- (y = w -> ( F ` y) = ( F ` w))
15	14	eleq1d 2103	. . . . . . . . . . . . . . . 16 |- (y = w -> (( F ` y) e. B <-> ( F ` w) e. B))
16	15	elrab 2692	. . . . . . . . . . . . . . 15 |- (w e. {y e. (A i^i dom F) | ( F ` y) e. B } <-> (w e. (A i^i dom F) /\ ( F ` w) e. B))
17	13, 16	syl6ibr 151	. . . . . . . . . . . . . 14 |- (z e. B -> ((w e. (A i^i dom F) /\ ( F ` w) = z) -> w e. {y e. (A i^i dom F) | ( F ` y) e. B }))
18		simpr 103	. . . . . . . . . . . . . . 15 |- ((w e. (A i^i dom F) /\ ( F ` w) = z) -> ( F ` w) = z)
19	18	a1i 9	. . . . . . . . . . . . . 14 |- (z e. B -> ((w e. (A i^i dom F) /\ ( F ` w) = z) -> ( F ` w) = z))
20	17, 19	jcad 291	. . . . . . . . . . . . 13 |- (z e. B -> ((w e. (A i^i dom F) /\ ( F ` w) = z) -> (w e. {y e. (A i^i dom F) | ( F ` y) e. B } /\ ( F ` w) = z)))
21	20	reximdv2 2412	. . . . . . . . . . . 12 |- (z e. B -> (E.w e. (A i^i dom F)( F ` w) = z -> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
22	21	adantl 262	. . . . . . . . . . 11 |- (( Fun F /\ z e. B) -> (E.w e. (A i^i dom F)( F ` w) = z -> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
23		funfn 4874	. . . . . . . . . . . . 13 |- ( Fun F <-> F Fn dom F)
24		inss2 3152	. . . . . . . . . . . . . . 15 |- (A i^i dom F) C_ dom F
25	6, 24	sstri 2948	. . . . . . . . . . . . . 14 |- {y e. (A i^i dom F) | ( F ` y) e. B } C_ dom F
26		fvelimab 5172	. . . . . . . . . . . . . 14 |- (( F Fn dom F /\ {y e. (A i^i dom F) | ( F ` y) e. B } C_ dom F) -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) <-> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
27	25, 26	mpan2 401	. . . . . . . . . . . . 13 |- ( F Fn dom F -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) <-> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
28	23, 27	sylbi 114	. . . . . . . . . . . 12 |- ( Fun F -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) <-> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
29	28	adantr 261	. . . . . . . . . . 11 |- (( Fun F /\ z e. B) -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) <-> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
30	22, 29	sylibrd 158	. . . . . . . . . 10 |- (( Fun F /\ z e. B) -> (E.w e. (A i^i dom F)( F ` w) = z -> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
31	11, 30	syld 40	. . . . . . . . 9 |- (( Fun F /\ z e. B) -> (z e. ( F "(A i^i dom F)) -> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
32	31	adantlr 446	. . . . . . . 8 |- ((( Fun F /\ B C_ ( F "(A i^i dom F))) /\ z e. B) -> (z e. ( F "(A i^i dom F)) -> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
33	8, 32	mpd 13	. . . . . . 7 |- ((( Fun F /\ B C_ ( F "(A i^i dom F))) /\ z e. B) -> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }))
34	33	ex 108	. . . . . 6 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> (z e. B -> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
35		fvelima 5168	. . . . . . . . 9 |- (( Fun F /\ z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })) -> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z)
36	35	ex 108	. . . . . . . 8 |- ( Fun F -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) -> E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z))
37		eleq1 2097	. . . . . . . . . . . 12 |- (( F ` w) = z -> (( F ` w) e. B <-> z e. B))
38	37	biimpcd 148	. . . . . . . . . . 11 |- (( F ` w) e. B -> (( F ` w) = z -> z e. B))
39	38	adantl 262	. . . . . . . . . 10 |- ((w e. (A i^i dom F) /\ ( F ` w) e. B) -> (( F ` w) = z -> z e. B))
40	16, 39	sylbi 114	. . . . . . . . 9 |- (w e. {y e. (A i^i dom F) | ( F ` y) e. B } -> (( F ` w) = z -> z e. B))
41	40	rexlimiv 2421	. . . . . . . 8 |- (E.w e. {y e. (A i^i dom F) | ( F ` y) e. B } ( F ` w) = z -> z e. B)
42	36, 41	syl6 29	. . . . . . 7 |- ( Fun F -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) -> z e. B))
43	42	adantr 261	. . . . . 6 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> (z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B }) -> z e. B))
44	34, 43	impbid 120	. . . . 5 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> (z e. B <-> z e. ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
45	44	eqrdv 2035	. . . 4 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> B = ( F " {y e. (A i^i dom F) | ( F ` y) e. B }))
46		ssimaex.1	. . . . . . 7 |- A e. _V
47	46	inex1 3882	. . . . . 6 |- (A i^i dom F) e. _V
48	47	rabex 3892	. . . . 5 |- {y e. (A i^i dom F) | ( F ` y) e. B } e. _V
49		sseq1 2960	. . . . . 6 |- (x = {y e. (A i^i dom F) | ( F ` y) e. B } -> (x C_ (A i^i dom F) <-> {y e. (A i^i dom F) | ( F ` y) e. B } C_ (A i^i dom F)))
50		imaeq2 4607	. . . . . . 7 |- (x = {y e. (A i^i dom F) | ( F ` y) e. B } -> ( F "x) = ( F " {y e. (A i^i dom F) | ( F ` y) e. B }))
51	50	eqeq2d 2048	. . . . . 6 |- (x = {y e. (A i^i dom F) | ( F ` y) e. B } -> (B = ( F "x) <-> B = ( F " {y e. (A i^i dom F) | ( F ` y) e. B })))
52	49, 51	anbi12d 442	. . . . 5 |- (x = {y e. (A i^i dom F) | ( F ` y) e. B } -> ((x C_ (A i^i dom F) /\ B = ( F "x)) <-> ( {y e. (A i^i dom F) | ( F ` y) e. B } C_ (A i^i dom F) /\ B = ( F " {y e. (A i^i dom F) | ( F ` y) e. B }))))
53	48, 52	spcev 2641	. . . 4 |- (( {y e. (A i^i dom F) | ( F ` y) e. B } C_ (A i^i dom F) /\ B = ( F " {y e. (A i^i dom F) | ( F ` y) e. B })) -> E.x(x C_ (A i^i dom F) /\ B = ( F "x)))
54	6, 45, 53	sylancr 393	. . 3 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> E.x(x C_ (A i^i dom F) /\ B = ( F "x)))
55		inss1 3151	. . . . . 6 |- (A i^i dom F) C_ A
56		sstr 2947	. . . . . 6 |- ((x C_ (A i^i dom F) /\ (A i^i dom F) C_ A) -> x C_ A)
57	55, 56	mpan2 401	. . . . 5 |- (x C_ (A i^i dom F) -> x C_ A)
58	57	anim1i 323	. . . 4 |- ((x C_ (A i^i dom F) /\ B = ( F "x)) -> (x C_ A /\ B = ( F "x)))
59	58	eximi 1488	. . 3 |- (E.x(x C_ (A i^i dom F) /\ B = ( F "x)) -> E.x(x C_ A /\ B = ( F "x)))
60	54, 59	syl 14	. 2 |- (( Fun F /\ B C_ ( F "(A i^i dom F))) -> E.x(x C_ A /\ B = ( F "x)))
61	5, 60	sylan2br 272	1 |- (( Fun F /\ B C_ ( F "A)) -> E.x(x C_ A /\ B = ( F "x)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  E.wrex 2301   {crab 2304   _Vcvv 2551   i^i cin 2910   C_ wss 2911   dom cdm 4288   |` cres 4290   "cima 4291   Fun wfun 4839   Fn wfn 4840   ` cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  ssimaexg  5178