Theorem ssimaex 6968

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 6013	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
2	1	imaeq2i 6062	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3		imadmres 6237	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ 𝐴)
4	2, 3	eqtr3i 2788	. . 3 ⊢ (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹 “ 𝐴)
5	4	sseq2i 3967	. 2 ⊢ (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹 “ 𝐴))
6		ssrab2 4035	. . . 4 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7		ssel2 3933	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
8	7	adantll 726	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)))
9		fvelima 6948	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧)
10	9	ex 417	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
11	10	adantr 485	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → ∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧))
12		eleq1a 2858	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → (𝐹‘𝑤) ∈ 𝐵))
13	12	anim2d 623	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵)))
14		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
15	14	eleq1d 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ 𝐵 ↔ (𝐹‘𝑤) ∈ 𝐵))
16	15	elrab 3651	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ↔ (𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵))
17	13, 16	imbitrrdi 255	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → 𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
18		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝐹‘𝑤) = 𝑧)
19	17, 18	jca2 522	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) = 𝑧) → (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∧ (𝐹‘𝑤) = 𝑧)))
20	19	reximdv2 3175	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
21	20	adantl 486	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
22		funfn 6568	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
23		inss2 4191	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
24	6, 23	sstri 3947	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ dom 𝐹
25		fvelimab 6955	. . . . . . . . . . . . . 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))
26	24, 25	mpan2 703	. . . . . . . . . . . . 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))
27	22, 26	sylbi 220	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) ↔ ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
28	27	adantr 485	. . . . . . . . . . 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))
29	21, 28	sylibrd 262	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (∃𝑤 ∈ (𝐴 ∩ dom 𝐹)(𝐹‘𝑤) = 𝑧 → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
30	11, 29	syld 48	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
31	30	adantlr 727	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → (𝑧 ∈ (𝐹 “ (𝐴 ∩ dom 𝐹)) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
32	8, 31	mpd 16	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
33	32	ex 417	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
34		fvelima 6948	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧)
35	34	ex 417	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → ∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧))
36		eleq1 2851	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑤) = 𝑧 → ((𝐹‘𝑤) ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
37	36	biimpcd 252	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
38	37	adantl 486	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (𝐴 ∩ dom 𝐹) ∧ (𝐹‘𝑤) ∈ 𝐵) → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
39	16, 38	sylbi 220	. . . . . . . . 9 ⊢ (𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → ((𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵))
40	39	rexlimiv 3159	. . . . . . . 8 ⊢ (∃𝑤 ∈ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} (𝐹‘𝑤) = 𝑧 → 𝑧 ∈ 𝐵)
41	35, 40	syl6 36	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → 𝑧 ∈ 𝐵))
42	41	adantr 485	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}) → 𝑧 ∈ 𝐵))
43	33, 42	impbid 215	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
44	43	eqrdv 2761	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
45		ssimaex.1	. . . . . . 7 ⊢ 𝐴 ∈ V
46	45	inex1 5287	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ∈ V
47	46	rabex 5311	. . . . 5 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∈ V
48		sseq1 3963	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝑥 ⊆ (𝐴 ∩ dom 𝐹) ↔ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)))
49		imaeq2 6060	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐹 “ 𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
50	49	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐵 = (𝐹 “ 𝑥) ↔ 𝐵 = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵})))
51	48, 50	anbi12d 643	. . . . 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}))))
52	47, 51	spcev 3566	. . . 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)))
53	6, 44, 52	sylancr 598	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)))
54		inss1 4190	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
55		sstr 3946	. . . . . 6 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ (𝐴 ∩ dom 𝐹) ⊆ 𝐴) → 𝑥 ⊆ 𝐴)
56	54, 55	mpan2 703	. . . . 5 ⊢ (𝑥 ⊆ (𝐴 ∩ dom 𝐹) → 𝑥 ⊆ 𝐴)
57	56	anim1i 626	. . . 4 ⊢ ((𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
58	57	eximi 1865	. . 3 ⊢ (∃𝑥(𝑥 ⊆ (𝐴 ∩ dom 𝐹) ∧ 𝐵 = (𝐹 “ 𝑥)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
59	53, 58	syl 18	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹))) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
60	5, 59	sylan2br 606	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  dom cdm 5663   ↾ cres 5665   “ cima 5666  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  ssimaexg  6969