Theorem fbasrn 22489

Description: Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
fbasrn.c	⊢ 𝐶 = ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))

Assertion

Ref	Expression
fbasrn	⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ∈ (fBas‘𝑌))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fbasrn

Dummy variables 𝑠 𝑟 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fbasrn.c	. . 3 ⊢ 𝐶 = ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))
2		simpl3 1190	. . . . . 6 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → 𝑌 ∈ 𝑉)
3		simpl2 1189	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑋⟶𝑌)
4		fimass 6529	. . . . . . 7 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 “ 𝑥) ⊆ 𝑌)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ⊆ 𝑌)
6	2, 5	sselpwd 5194	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ 𝒫 𝑌)
7	6	fmpttd 6856	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)):𝐵⟶𝒫 𝑌)
8	7	frnd 6494	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ⊆ 𝒫 𝑌)
9	1, 8	eqsstrid 3963	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ⊆ 𝒫 𝑌)
10	1	a1i 11	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 = ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
11		ffun 6490	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
12	11	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → Fun 𝐹)
13		funimaexg 6410	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ V)
14	13	ralrimiva 3149	. . . . . . 7 ⊢ (Fun 𝐹 → ∀𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ∈ V)
15		dmmptg 6063	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ∈ V → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = 𝐵)
16	12, 14, 15	3syl 18	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = 𝐵)
17		fbasne0 22435	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ≠ ∅)
18	17	3ad2ant1 1130	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐵 ≠ ∅)
19	16, 18	eqnetrd 3054	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
20		dm0rn0 5759	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = ∅ ↔ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = ∅)
21	20	necon3bii 3039	. . . . 5 ⊢ (dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅ ↔ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
22	19, 21	sylib 221	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
23	10, 22	eqnetrd 3054	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ≠ ∅)
24		fbelss 22438	. . . . . . . . 9 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑥 ⊆ 𝑋)
25	24	ex 416	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝑋))
26	25	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝑋))
27		0nelfb 22436	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐵)
28		eleq1 2877	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐵 ↔ ∅ ∈ 𝐵))
29	28	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝐵 ↔ ¬ ∅ ∈ 𝐵))
30	27, 29	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 = ∅ → ¬ 𝑥 ∈ 𝐵))
31	30	con2d 136	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐵 → ¬ 𝑥 = ∅))
32	31	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → ¬ 𝑥 = ∅))
33	26, 32	jcad 516	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → (𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅)))
34		fdm 6495	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
35	34	3ad2ant2 1131	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom 𝐹 = 𝑋)
36	35	sseq2d 3947	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ⊆ dom 𝐹 ↔ 𝑥 ⊆ 𝑋))
37	36	biimpar 481	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ dom 𝐹)
38		sseqin2 4142	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑥) = 𝑥)
39	37, 38	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → (dom 𝐹 ∩ 𝑥) = 𝑥)
40	39	eqeq1d 2800	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → ((dom 𝐹 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
41	40	biimpd 232	. . . . . . . . 9 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → ((dom 𝐹 ∩ 𝑥) = ∅ → 𝑥 = ∅))
42	41	con3d 155	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 = ∅ → ¬ (dom 𝐹 ∩ 𝑥) = ∅))
43	42	expimpd 457	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅) → ¬ (dom 𝐹 ∩ 𝑥) = ∅))
44		eqcom 2805	. . . . . . . . 9 ⊢ (∅ = (𝐹 “ 𝑥) ↔ (𝐹 “ 𝑥) = ∅)
45		imadisj 5915	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = ∅ ↔ (dom 𝐹 ∩ 𝑥) = ∅)
46	44, 45	bitri 278	. . . . . . . 8 ⊢ (∅ = (𝐹 “ 𝑥) ↔ (dom 𝐹 ∩ 𝑥) = ∅)
47	46	notbii 323	. . . . . . 7 ⊢ (¬ ∅ = (𝐹 “ 𝑥) ↔ ¬ (dom 𝐹 ∩ 𝑥) = ∅)
48	43, 47	syl6ibr 255	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅) → ¬ ∅ = (𝐹 “ 𝑥)))
49	33, 48	syld 47	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → ¬ ∅ = (𝐹 “ 𝑥)))
50	49	ralrimiv 3148	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥))
51	1	eleq2i 2881	. . . . . . 7 ⊢ (∅ ∈ 𝐶 ↔ ∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
52		0ex 5175	. . . . . . . 8 ⊢ ∅ ∈ V
53		eqid 2798	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))
54	53	elrnmpt 5792	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥)))
55	52, 54	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
56	51, 55	bitri 278	. . . . . 6 ⊢ (∅ ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
57	56	notbii 323	. . . . 5 ⊢ (¬ ∅ ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
58		df-nel 3092	. . . . 5 ⊢ (∅ ∉ 𝐶 ↔ ¬ ∅ ∈ 𝐶)
59		ralnex 3199	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
60	57, 58, 59	3bitr4i 306	. . . 4 ⊢ (∅ ∉ 𝐶 ↔ ∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥))
61	50, 60	sylibr 237	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∅ ∉ 𝐶)
62	1	eleq2i 2881	. . . . . . . 8 ⊢ (𝑟 ∈ 𝐶 ↔ 𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
63		imaeq2 5892	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹 “ 𝑥) = (𝐹 “ 𝑢))
64	63	cbvmptv 5133	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑢 ∈ 𝐵 ↦ (𝐹 “ 𝑢))
65	64	elrnmpt 5792	. . . . . . . . 9 ⊢ (𝑟 ∈ V → (𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢)))
66	65	elv 3446	. . . . . . . 8 ⊢ (𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢))
67	62, 66	bitri 278	. . . . . . 7 ⊢ (𝑟 ∈ 𝐶 ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢))
68	1	eleq2i 2881	. . . . . . . 8 ⊢ (𝑠 ∈ 𝐶 ↔ 𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
69		imaeq2 5892	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝐹 “ 𝑥) = (𝐹 “ 𝑣))
70	69	cbvmptv 5133	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑣 ∈ 𝐵 ↦ (𝐹 “ 𝑣))
71	70	elrnmpt 5792	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
72	71	elv 3446	. . . . . . . 8 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣))
73	68, 72	bitri 278	. . . . . . 7 ⊢ (𝑠 ∈ 𝐶 ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣))
74	67, 73	anbi12i 629	. . . . . 6 ⊢ ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) ↔ (∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
75		reeanv 3320	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) ↔ (∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
76	74, 75	bitr4i 281	. . . . 5 ⊢ ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) ↔ ∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))
77		fbasssin 22441	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
78	77	3expb 1117	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
79	78	3ad2antl1 1182	. . . . . . . . 9 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
80	79	adantrr 716	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
81		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑤) = (𝐹 “ 𝑤)
82		imaeq2 5892	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹 “ 𝑥) = (𝐹 “ 𝑤))
83	82	rspceeqv 3586	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝐵 ∧ (𝐹 “ 𝑤) = (𝐹 “ 𝑤)) → ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥))
84	81, 83	mpan2 690	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥))
85	84	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥))
86	1	eleq2i 2881	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑤) ∈ 𝐶 ↔ (𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
87		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
88	87	funimaex 6411	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (𝐹 “ 𝑤) ∈ V)
89	53	elrnmpt 5792	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑤) ∈ V → ((𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
90	12, 88, 89	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
91	86, 90	syl5bb 286	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝐹 “ 𝑤) ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
92	91	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → ((𝐹 “ 𝑤) ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
93	85, 92	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ 𝑤) ∈ 𝐶)
94		imass2 5932	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ (𝑢 ∩ 𝑣) → (𝐹 “ 𝑤) ⊆ (𝐹 “ (𝑢 ∩ 𝑣)))
95	94	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ (𝑢 ∩ 𝑣)))
96		inss1 4155	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
97		imass2 5932	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑢 → (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑢))
98	96, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑢)
99		inss2 4156	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑣
100		imass2 5932	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑣 → (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑣))
101	99, 100	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑣)
102	98, 101	ssini 4158	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ ((𝐹 “ 𝑢) ∩ (𝐹 “ 𝑣))
103		ineq12 4134	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → (𝑟 ∩ 𝑠) = ((𝐹 “ 𝑢) ∩ (𝐹 “ 𝑣)))
104	103	ad2antlr 726	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝑟 ∩ 𝑠) = ((𝐹 “ 𝑢) ∩ (𝐹 “ 𝑣)))
105	102, 104	sseqtrrid 3968	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝑟 ∩ 𝑠))
106	95, 105	sstrd 3925	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠))
107		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹 “ 𝑤) → (𝑧 ⊆ (𝑟 ∩ 𝑠) ↔ (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠)))
108	107	rspcev 3571	. . . . . . . . . 10 ⊢ (((𝐹 “ 𝑤) ∈ 𝐶 ∧ (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
109	93, 106, 108	syl2anc 587	. . . . . . . . 9 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
110	109	adantlrl 719	. . . . . . . 8 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
111	80, 110	rexlimddv 3250	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
112	111	exp32 424	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ((𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))))
113	112	rexlimdvv 3252	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
114	76, 113	syl5bi 245	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
115	114	ralrimivv 3155	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
116	23, 61, 115	3jca 1125	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
117		isfbas2 22440	. . 3 ⊢ (𝑌 ∈ 𝑉 → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))))
118	117	3ad2ant3 1132	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))))
119	9, 116, 118	mpbir2and 712	1 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ∈ (fBas‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∉ wnel 3091  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   “ cima 5522  Fun wfun 6318  ⟶wf 6320  ‘cfv 6324  fBascfbas 20079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-fbas 20088

This theorem is referenced by:  fmfil  22549  fmss  22551  elfm  22552  fmucnd  22898  fmcfil  23876