Theorem fbasrn 21688

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		simpl2 1065	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑋⟶𝑌)
3		imassrn 5477	. . . . . . . 8 ⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹
4		frn 6053	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → ran 𝐹 ⊆ 𝑌)
5	3, 4	syl5ss 3614	. . . . . . 7 ⊢ (𝐹:𝑋⟶𝑌 → (𝐹 “ 𝑥) ⊆ 𝑌)
6	2, 5	syl 17	. . . . . 6 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ⊆ 𝑌)
7		simpl3 1066	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → 𝑌 ∈ 𝑉)
8		elpw2g 4827	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ((𝐹 “ 𝑥) ∈ 𝒫 𝑌 ↔ (𝐹 “ 𝑥) ⊆ 𝑌))
9	7, 8	syl 17	. . . . . 6 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → ((𝐹 “ 𝑥) ∈ 𝒫 𝑌 ↔ (𝐹 “ 𝑥) ⊆ 𝑌))
10	6, 9	mpbird 247	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ 𝒫 𝑌)
11		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))
12	10, 11	fmptd 6385	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)):𝐵⟶𝒫 𝑌)
13		frn 6053	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)):𝐵⟶𝒫 𝑌 → ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ⊆ 𝒫 𝑌)
14	12, 13	syl 17	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ⊆ 𝒫 𝑌)
15	1, 14	syl5eqss 3649	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ⊆ 𝒫 𝑌)
16	1	a1i 11	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 = ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
17		ffun 6048	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
18	17	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → Fun 𝐹)
19		funimaexg 5975	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ 𝑥) ∈ V)
20	19	ralrimiva 2966	. . . . . . 7 ⊢ (Fun 𝐹 → ∀𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ∈ V)
21		dmmptg 5632	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ∈ V → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = 𝐵)
22	18, 20, 21	3syl 18	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = 𝐵)
23		fbasne0 21634	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ≠ ∅)
24	23	3ad2ant1 1082	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐵 ≠ ∅)
25	22, 24	eqnetrd 2861	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
26		dm0rn0 5342	. . . . . 6 ⊢ (dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = ∅ ↔ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = ∅)
27	26	necon3bii 2846	. . . . 5 ⊢ (dom (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅ ↔ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
28	25, 27	sylib 208	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ≠ ∅)
29	16, 28	eqnetrd 2861	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ≠ ∅)
30		fbelss 21637	. . . . . . . . 9 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ 𝐵) → 𝑥 ⊆ 𝑋)
31	30	ex 450	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝑋))
32	31	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝑋))
33		0nelfb 21635	. . . . . . . . . 10 ⊢ (𝐵 ∈ (fBas‘𝑋) → ¬ ∅ ∈ 𝐵)
34		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐵 ↔ ∅ ∈ 𝐵))
35	34	notbid 308	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ 𝐵 ↔ ¬ ∅ ∈ 𝐵))
36	33, 35	syl5ibrcom 237	. . . . . . . . 9 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 = ∅ → ¬ 𝑥 ∈ 𝐵))
37	36	con2d 129	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑥 ∈ 𝐵 → ¬ 𝑥 = ∅))
38	37	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → ¬ 𝑥 = ∅))
39	32, 38	jcad 555	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → (𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅)))
40		fdm 6051	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
41	40	3ad2ant2 1083	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → dom 𝐹 = 𝑋)
42	41	sseq2d 3633	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ⊆ dom 𝐹 ↔ 𝑥 ⊆ 𝑋))
43	42	biimpar 502	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ dom 𝐹)
44		sseqin2 3817	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑥) = 𝑥)
45	43, 44	sylib 208	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → (dom 𝐹 ∩ 𝑥) = 𝑥)
46	45	eqeq1d 2624	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → ((dom 𝐹 ∩ 𝑥) = ∅ ↔ 𝑥 = ∅))
47	46	biimpd 219	. . . . . . . . 9 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → ((dom 𝐹 ∩ 𝑥) = ∅ → 𝑥 = ∅))
48	47	con3d 148	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ⊆ 𝑋) → (¬ 𝑥 = ∅ → ¬ (dom 𝐹 ∩ 𝑥) = ∅))
49	48	expimpd 629	. . . . . . 7 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅) → ¬ (dom 𝐹 ∩ 𝑥) = ∅))
50		eqcom 2629	. . . . . . . . 9 ⊢ (∅ = (𝐹 “ 𝑥) ↔ (𝐹 “ 𝑥) = ∅)
51		imadisj 5484	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = ∅ ↔ (dom 𝐹 ∩ 𝑥) = ∅)
52	50, 51	bitri 264	. . . . . . . 8 ⊢ (∅ = (𝐹 “ 𝑥) ↔ (dom 𝐹 ∩ 𝑥) = ∅)
53	52	notbii 310	. . . . . . 7 ⊢ (¬ ∅ = (𝐹 “ 𝑥) ↔ ¬ (dom 𝐹 ∩ 𝑥) = ∅)
54	49, 53	syl6ibr 242	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑥 ⊆ 𝑋 ∧ ¬ 𝑥 = ∅) → ¬ ∅ = (𝐹 “ 𝑥)))
55	39, 54	syld 47	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝑥 ∈ 𝐵 → ¬ ∅ = (𝐹 “ 𝑥)))
56	55	ralrimiv 2965	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥))
57	1	eleq2i 2693	. . . . . . 7 ⊢ (∅ ∈ 𝐶 ↔ ∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
58		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
59	11	elrnmpt 5372	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥)))
60	58, 59	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
61	57, 60	bitri 264	. . . . . 6 ⊢ (∅ ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
62	61	notbii 310	. . . . 5 ⊢ (¬ ∅ ∈ 𝐶 ↔ ¬ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
63		df-nel 2898	. . . . 5 ⊢ (∅ ∉ 𝐶 ↔ ¬ ∅ ∈ 𝐶)
64		ralnex 2992	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥) ↔ ¬ ∃𝑥 ∈ 𝐵 ∅ = (𝐹 “ 𝑥))
65	62, 63, 64	3bitr4i 292	. . . 4 ⊢ (∅ ∉ 𝐶 ↔ ∀𝑥 ∈ 𝐵 ¬ ∅ = (𝐹 “ 𝑥))
66	56, 65	sylibr 224	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∅ ∉ 𝐶)
67	1	eleq2i 2693	. . . . . . . 8 ⊢ (𝑟 ∈ 𝐶 ↔ 𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
68		vex 3203	. . . . . . . . 9 ⊢ 𝑟 ∈ V
69		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹 “ 𝑥) = (𝐹 “ 𝑢))
70	69	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑢 ∈ 𝐵 ↦ (𝐹 “ 𝑢))
71	70	elrnmpt 5372	. . . . . . . . 9 ⊢ (𝑟 ∈ V → (𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢)))
72	68, 71	ax-mp 5	. . . . . . . 8 ⊢ (𝑟 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢))
73	67, 72	bitri 264	. . . . . . 7 ⊢ (𝑟 ∈ 𝐶 ↔ ∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢))
74	1	eleq2i 2693	. . . . . . . 8 ⊢ (𝑠 ∈ 𝐶 ↔ 𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
75		vex 3203	. . . . . . . . 9 ⊢ 𝑠 ∈ V
76		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝐹 “ 𝑥) = (𝐹 “ 𝑣))
77	76	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) = (𝑣 ∈ 𝐵 ↦ (𝐹 “ 𝑣))
78	77	elrnmpt 5372	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
79	75, 78	ax-mp 5	. . . . . . . 8 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣))
80	74, 79	bitri 264	. . . . . . 7 ⊢ (𝑠 ∈ 𝐶 ↔ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣))
81	73, 80	anbi12i 733	. . . . . 6 ⊢ ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) ↔ (∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
82		reeanv 3107	. . . . . 6 ⊢ (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) ↔ (∃𝑢 ∈ 𝐵 𝑟 = (𝐹 “ 𝑢) ∧ ∃𝑣 ∈ 𝐵 𝑠 = (𝐹 “ 𝑣)))
83	81, 82	bitr4i 267	. . . . 5 ⊢ ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) ↔ ∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))
84		fbasssin 21640	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
85	84	3expb 1266	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
86	85	3ad2antl1 1223	. . . . . . . . 9 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
87	86	adantrr 753	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ (𝑢 ∩ 𝑣))
88		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝐹 “ 𝑤) = (𝐹 “ 𝑤)
89		imaeq2 5462	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐹 “ 𝑥) = (𝐹 “ 𝑤))
90	89	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ((𝐹 “ 𝑤) = (𝐹 “ 𝑥) ↔ (𝐹 “ 𝑤) = (𝐹 “ 𝑤)))
91	90	rspcev 3309	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝐵 ∧ (𝐹 “ 𝑤) = (𝐹 “ 𝑤)) → ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥))
92	88, 91	mpan2 707	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥))
93	92	ad2antrl 764	. . . . . . . . . . 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))
94	1	eleq2i 2693	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑤) ∈ 𝐶 ↔ (𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)))
95		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
96	95	funimaex 5976	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (𝐹 “ 𝑤) ∈ V)
97	11	elrnmpt 5372	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑤) ∈ V → ((𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
98	18, 96, 97	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝐹 “ 𝑤) ∈ ran (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
99	94, 98	syl5bb 272	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝐹 “ 𝑤) ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑤) = (𝐹 “ 𝑥)))
100	99	ad2antrr 762	. . . . . . . . . . 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)))
101	93, 100	mpbird 247	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ 𝑤) ∈ 𝐶)
102		imass2 5501	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ (𝑢 ∩ 𝑣) → (𝐹 “ 𝑤) ⊆ (𝐹 “ (𝑢 ∩ 𝑣)))
103	102	ad2antll 765	. . . . . . . . . . 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)))
104		inss1 3833	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
105		imass2 5501	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑢 → (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑢))
106	104, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑢)
107		inss2 3834	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑣
108		imass2 5501	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑣 → (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑣))
109	107, 108	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ (𝐹 “ 𝑣)
110	106, 109	ssini 3836	. . . . . . . . . . . 12 ⊢ (𝐹 “ (𝑢 ∩ 𝑣)) ⊆ ((𝐹 “ 𝑢) ∩ (𝐹 “ 𝑣))
111		ineq12 3809	. . . . . . . . . . . . 13 ⊢ ((𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → (𝑟 ∩ 𝑠) = ((𝐹 “ 𝑢) ∩ (𝐹 “ 𝑣)))
112	111	ad2antlr 763	. . . . . . . . . . . 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)))
113	110, 112	syl5sseqr 3654	. . . . . . . . . . 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))
114	103, 113	sstrd 3613	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣))) ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑢 ∩ 𝑣))) → (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠))
115		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹 “ 𝑤) → (𝑧 ⊆ (𝑟 ∩ 𝑠) ↔ (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠)))
116	115	rspcev 3309	. . . . . . . . . 10 ⊢ (((𝐹 “ 𝑤) ∈ 𝐶 ∧ (𝐹 “ 𝑤) ⊆ (𝑟 ∩ 𝑠)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
117	101, 114, 116	syl2anc 693	. . . . . . . . 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))
118	117	adantlrl 756	. . . . . . . 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))
119	87, 118	rexlimddv 3035	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)))) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
120	119	exp32 631	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → ((𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))))
121	120	rexlimdvv 3037	. . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (∃𝑢 ∈ 𝐵 ∃𝑣 ∈ 𝐵 (𝑟 = (𝐹 “ 𝑢) ∧ 𝑠 = (𝐹 “ 𝑣)) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
122	83, 121	syl5bi 232	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ((𝑟 ∈ 𝐶 ∧ 𝑠 ∈ 𝐶) → ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
123	122	ralrimivv 2970	. . 3 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠))
124	29, 66, 123	3jca 1242	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))
125		isfbas2 21639	. . 3 ⊢ (𝑌 ∈ 𝑉 → (𝐶 ∈ (fBas‘𝑌) ↔ (𝐶 ⊆ 𝒫 𝑌 ∧ (𝐶 ≠ ∅ ∧ ∅ ∉ 𝐶 ∧ ∀𝑟 ∈ 𝐶 ∀𝑠 ∈ 𝐶 ∃𝑧 ∈ 𝐶 𝑧 ⊆ (𝑟 ∩ 𝑠)))))
126	125	3ad2ant3 1084	. 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)))))
127	15, 124, 126	mpbir2and 957	1 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ∈ (fBas‘𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∉ wnel 2897  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   “ cima 5117  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  fBascfbas 19734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-fbas 19743

This theorem is referenced by:  fmfil  21748  fmss  21750  elfm  21751  fmucnd  22096  fmcfil  23070