Theorem ssimaex 5737

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

Hypothesis

Ref	Expression
ssimaex.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ssimaex	⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 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 5058	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
2	1	imaeq2i 5098	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3		imadmres 5254	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ 𝐴)
4	2, 3	eqtr3i 2255	. . 3 ⊢ (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹 “ 𝐴)
5	4	sseq2i 3264	. 2 ⊢ (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹 “ 𝐴))
6		ssrab2 3322	. . . 4 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7		ssel2 3232	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
8	7	adantll 476	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9		fvelima 5727	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧)
10	9	ex 115	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
11	10	adantr 276	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
12		eleq1a 2304	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → (𝐹‘𝑤) ∈ 𝐵))
13	12	anim2d 337	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵)))
14		fveq2 5669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
15	14	eleq1d 2301	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ 𝐵 ↔ (𝐹‘𝑤) ∈ 𝐵))
16	15	elrab 2972	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵))
17	13, 16	imbitrrdi 162	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
18		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑧)
19	18	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑧))
20	17, 19	jcad 307	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∧ (𝐹‘𝑤) = 𝑧)))
21	20	reximdv2 2641	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
22	21	adantl 277	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
23		funfn 5381	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
24		inss2 3441	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
25	6, 24	sstri 3246	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ dom 𝐹
26		fvelimab 5732	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn dom 𝐹 ∧ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ dom 𝐹) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
27	25, 26	mpan2 425	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn dom 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
28	23, 27	sylbi 121	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
29	28	adantr 276	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
30	22, 29	sylibrd 169	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
31	11, 30	syld 45	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
32	31	adantlr 477	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
33	8, 32	mpd 13	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
34	33	ex 115	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
35		fvelima 5727	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧)
36	35	ex 115	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
37		eleq1 2295	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑤) = 𝑧 → ((𝐹‘𝑤) ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
38	37	biimpcd 159	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
39	38	adantl 277	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵) → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
40	16, 39	sylbi 121	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
41	40	rexlimiv 2654	. . . . . . . 8 ⊢ (∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵)
42	36, 41	syl6 33	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → 𝑧 ∈ 𝐵))
43	42	adantr 276	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → 𝑧 ∈ 𝐵))
44	34, 43	impbid 129	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
45	44	eqrdv 2230	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
46		ssimaex.1	. . . . . . 7 ⊢ 𝐴 ∈ V
47	46	inex1 4243	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ∈ V
48	47	rabex 4255	. . . . 5 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∈ V
49		sseq1 3260	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
50		imaeq2 5096	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐹 “ 𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
51	50	eqeq2d 2244	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐵 = (𝐹 “ 𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
52	49, 51	anbi12d 473	. . . . 5 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) ↔ ({𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))))
53	48, 52	spcev 2911	. . . 4 ⊢ (({𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)))
54	6, 45, 53	sylancr 414	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)))
55		inss1 3440	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
56		sstr 3245	. . . . . 6 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
57	55, 56	mpan2 425	. . . . 5 ⊢ (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥 ⊆ 𝐴)
58	57	anim1i 340	. . . 4 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
59	58	eximi 1649	. . 3 ⊢ (∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
60	54, 59	syl 14	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
61	5, 60	sylan2br 288	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  {crab 2524  Vcvv 2812   ∩ cin 3209   ⊆ wss 3210  dom cdm 4748   ↾ cres 4750   “ cima 4751  Fun wfun 5345   Fn wfn 5346  ‘cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359

This theorem is referenced by:  ssimaexg  5738