Theorem ssimaex 5255

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 4650	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
2	1	imaeq2i 4686	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3		imadmres 4833	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ 𝐴)
4	2, 3	eqtr3i 2103	. . 3 ⊢ (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹 “ 𝐴)
5	4	sseq2i 3024	. 2 ⊢ (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹 “ 𝐴))
6		ssrab2 3079	. . . 4 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7		ssel2 2994	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
8	7	adantll 459	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9		fvelima 5246	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧)
10	9	ex 113	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
11	10	adantr 270	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
12		eleq1a 2150	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → (𝐹‘𝑤) ∈ 𝐵))
13	12	anim2d 330	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵)))
14		fveq2 5198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
15	14	eleq1d 2147	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ 𝐵 ↔ (𝐹‘𝑤) ∈ 𝐵))
16	15	elrab 2749	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵))
17	13, 16	syl6ibr 160	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
18		simpr 108	. . . . . . . . . . . . . . 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 301	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∧ (𝐹‘𝑤) = 𝑧)))
21	20	reximdv2 2460	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
22	21	adantl 271	. . . . . . . . . . 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 4951	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
24		inss2 3187	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
25	6, 24	sstri 3008	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ dom 𝐹
26		fvelimab 5250	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 119	. . . . . . . . . . . 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 270	. . . . . . . . . . 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 167	. . . . . . . . . 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 44	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
32	31	adantlr 460	. . . . . . . 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 113	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
35		fvelima 5246	. . . . . . . . 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 113	. . . . . . . 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 2141	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑤) = 𝑧 → ((𝐹‘𝑤) ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
38	37	biimpcd 157	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
39	38	adantl 271	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵) → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
40	16, 39	sylbi 119	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
41	40	rexlimiv 2471	. . . . . . . 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 270	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → 𝑧 ∈ 𝐵))
44	34, 43	impbid 127	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
45	44	eqrdv 2079	. . . 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 3912	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ∈ V
48	47	rabex 3922	. . . . 5 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∈ V
49		sseq1 3020	. . . . . 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 4684	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐹 “ 𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
51	50	eqeq2d 2092	. . . . . 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 456	. . . . 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 2692	. . . 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 405	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)))
55		inss1 3186	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
56		sstr 3007	. . . . . 6 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
57	55, 56	mpan2 415	. . . . 5 ⊢ (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥 ⊆ 𝐴)
58	57	anim1i 333	. . . 4 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
59	58	eximi 1531	. . 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 282	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  {crab 2352  Vcvv 2601   ∩ cin 2972   ⊆ wss 2973  dom cdm 4363   ↾ cres 4365   “ cima 4366  Fun wfun 4916   Fn wfn 4917  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  ssimaexg  5256