Theorem ssimaex 4729

Description: The existence of a subimage.

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

Step	Hyp	Ref	Expression
1		ssimaex.1	. . . . . . 7 |- A e. _V
2	1	inex1 3452	. . . . . 6 |- (A i^i dom F) e. _V
3	2	rabex 3461	. . . . 5 |- {y e. (A i^i dom F) | (F` y) e. B} e. _V
4		sseq1 2637	. . . . . 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)))
5		imaeq2 4260	. . . . . . 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}))
6	5	eqeq2d 1895	. . . . . 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})))
7	4, 6	anbi12d 690	. . . . 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}))))
8	3, 7	cla4ev 2371	. . . 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)))
9		ssrab2 2692	. . . 4 |- {y e. (A i^i dom F) | (F` y) e. B} C_ (A i^i dom F)
10		ssel2 2616	. . . . . . . . 9 |- ((B C_ (F"(A i^i dom F)) /\ z e. B) -> z e. (F"(A i^i dom F)))
11	10	adantll 428	. . . . . . . 8 |- (((Fun F /\ B C_ (F"(A i^i dom F))) /\ z e. B) -> z e. (F"(A i^i dom F)))
12		fvelima 4723	. . . . . . . . . . . 12 |- ((Fun F /\ z e. (F"(A i^i dom F))) -> E.w e. (A i^i dom F)(F` w) = z)
13	12	ex 402	. . . . . . . . . . 11 |- (Fun F -> (z e. (F"(A i^i dom F)) -> E.w e. (A i^i dom F)(F` w) = z))
14	13	adantr 425	. . . . . . . . . 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))
15		eleq1a 1966	. . . . . . . . . . . . . . . 16 |- (z e. B -> ((F` w) = z -> (F` w) e. B))
16	15	anim2d 620	. . . . . . . . . . . . . . 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)))
17		fveq2 4681	. . . . . . . . . . . . . . . . 17 |- (y = w -> (F` y) = (F` w))
18	17	eleq1d 1963	. . . . . . . . . . . . . . . 16 |- (y = w -> ((F` y) e. B <-> (F` w) e. B))
19	18	elrab 2414	. . . . . . . . . . . . . . 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))
20	16, 19	syl6ibr 230	. . . . . . . . . . . . . 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}))
21		simpr 350	. . . . . . . . . . . . . . 15 |- ((w e. (A i^i dom F) /\ (F` w) = z) -> (F` w) = z)
22	21	a1i 8	. . . . . . . . . . . . . 14 |- (z e. B -> ((w e. (A i^i dom F) /\ (F` w) = z) -> (F` w) = z))
23	20, 22	jcad 661	. . . . . . . . . . . . 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)))
24	23	reximdv2 2200	. . . . . . . . . . . 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))
25	24	adantl 424	. . . . . . . . . . 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))
26		funfn 4451	. . . . . . . . . . . . 13 |- (Fun F <-> F Fn dom F)
27		inss2 2813	. . . . . . . . . . . . . . 15 |- (A i^i dom F) C_ dom F
28	9, 27	sstri 2626	. . . . . . . . . . . . . 14 |- {y e. (A i^i dom F) | (F` y) e. B} C_ dom F
29		fvelimab 4725	. . . . . . . . . . . . . 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))
30	28, 29	mpan2 760	. . . . . . . . . . . . 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))
31	26, 30	sylbi 216	. . . . . . . . . . . 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))
32	31	adantr 425	. . . . . . . . . . 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))
33	25, 32	sylibrd 221	. . . . . . . . . 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})))
34	14, 33	syld 30	. . . . . . . . 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})))
35	34	adantlr 429	. . . . . . . 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})))
36	11, 35	mpd 29	. . . . . . 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}))
37	36	ex 402	. . . . . 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})))
38		fvelima 4723	. . . . . . . . 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)
39	38	ex 402	. . . . . . . 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))
40		eleq1 1957	. . . . . . . . . . . 12 |- ((F` w) = z -> ((F` w) e. B <-> z e. B))
41	40	biimpcd 172	. . . . . . . . . . 11 |- ((F` w) e. B -> ((F` w) = z -> z e. B))
42	41	adantl 424	. . . . . . . . . 10 |- ((w e. (A i^i dom F) /\ (F` w) e. B) -> ((F` w) = z -> z e. B))
43	19, 42	sylbi 216	. . . . . . . . 9 |- (w e. {y e. (A i^i dom F) | (F` y) e. B} -> ((F` w) = z -> z e. B))
44	43	r19.23aiv 2211	. . . . . . . 8 |- (E.w e. {y e. (A i^i dom F) | (F` y) e. B} (F` w) = z -> z e. B)
45	39, 44	syl6 25	. . . . . . 7 |- (Fun F -> (z e. (F"{y e. (A i^i dom F) | (F` y) e. B}) -> z e. B))
46	45	adantr 425	. . . . . 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))
47	37, 46	impbid 574	. . . . 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})))
48	47	eqrdv 1882	. . . 4 |- ((Fun F /\ B C_ (F"(A i^i dom F))) -> B = (F"{y e. (A i^i dom F) | (F` y) e. B}))
49	8, 9, 48	sylancr 526	. . 3 |- ((Fun F /\ B C_ (F"(A i^i dom F))) -> E.x(x C_ (A i^i dom F) /\ B = (F"x)))
50		inss1 2812	. . . . . 6 |- (A i^i dom F) C_ A
51		sstr 2625	. . . . . 6 |- ((x C_ (A i^i dom F) /\ (A i^i dom F) C_ A) -> x C_ A)
52	50, 51	mpan2 760	. . . . 5 |- (x C_ (A i^i dom F) -> x C_ A)
53	52	anim1i 361	. . . 4 |- ((x C_ (A i^i dom F) /\ B = (F"x)) -> (x C_ A /\ B = (F"x)))
54	53	eximi 1387	. . 3 |- (E.x(x C_ (A i^i dom F) /\ B = (F"x)) -> E.x(x C_ A /\ B = (F"x)))
55	49, 54	syl 12	. 2 |- ((Fun F /\ B C_ (F"(A i^i dom F))) -> E.x(x C_ A /\ B = (F"x)))
56		dmres 4234	. . . . 5 |- dom ( F |` A) = (A i^i dom F)
57	56	imaeq2i 4262	. . . 4 |- (F"dom ( F |` A)) = (F"(A i^i dom F))
58		imadmres 4391	. . . 4 |- (F"dom ( F |` A)) = (F"A)
59	57, 58	eqtr3i 1910	. . 3 |- (F"(A i^i dom F)) = (F"A)
60	59	sseq2i 2642	. 2 |- (B C_ (F"(A i^i dom F)) <-> B C_ (F"A))
61	55, 60	sylan2br 502	1 |- ((Fun F /\ B C_ (F"A)) -> E.x(x C_ A /\ B = (F"x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  {crab 2108  _Vcvv 2292   i^i cin 2592   C_ wss 2593  dom cdm 3986   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  ssimaexg 4730

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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