Theorem ssimaex 5622

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 4967	. . . . 5 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
2	1	imaeq2i 5007	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ (𝐴 ∩ dom 𝐹))
3		imadmres 5162	. . . 4 ⊢ (𝐹 “ dom (𝐹 ↾ 𝐴)) = (𝐹 “ 𝐴)
4	2, 3	eqtr3i 2219	. . 3 ⊢ (𝐹 “ (𝐴 ∩ dom 𝐹)) = (𝐹 “ 𝐴)
5	4	sseq2i 3210	. 2 ⊢ (𝐵 ⊆ (𝐹 “ (𝐴 ∩ dom 𝐹)) ↔ 𝐵 ⊆ (𝐹 “ 𝐴))
6		ssrab2 3268	. . . 4 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ (𝐴 ∩ dom 𝐹)
7		ssel2 3178	. . . . . . . . 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 5612	. . . . . . . . . . . 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 2268	. . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
15	14	eleq1d 2265	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ 𝐵 ↔ (𝐹‘𝑤) ∈ 𝐵))
16	15	elrab 2920	. . . . . . . . . . . . . . 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 2596	. . . . . . . . . . . 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 5288	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
24		inss2 3384	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ dom 𝐹) ⊆ dom 𝐹
25	6, 24	sstri 3192	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ⊆ dom 𝐹
26		fvelimab 5617	. . . . . . . . . . . . . 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 5612	. . . . . . . . 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 2259	. . . . . . . . . . . 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 2608	. . . . . . . 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 2194	. . . 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 4167	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ∈ V
48	47	rabex 4177	. . . . 5 ⊢ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} ∈ V
49		sseq1 3206	. . . . . 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 5005	. . . . . . 7 ⊢ (𝑥 = {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵} → (𝐹 “ 𝑥) = (𝐹 “ {𝑦 ∈ (𝐴 ∩ dom 𝐹) ∣ (𝐹‘𝑦) ∈ 𝐵}))
51	50	eqeq2d 2208	. . . . . 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 2859	. . . 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 3383	. . . . . 6 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
56		sstr 3191	. . . . . 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 1614	. . 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 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476  {crab 2479  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  dom cdm 4663   ↾ cres 4665   “ cima 4666  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  ssimaexg  5623