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Theorem fmfnfm 24020

Description: A filter finer than an image filter is an image filter of the same function. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

**Hypotheses**
Ref	Expression
fmfnfm.b	⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌))
fmfnfm.l	⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋))
fmfnfm.f	⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
fmfnfm.fm	⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)

**Assertion**
Ref	Expression
fmfnfm	⊢ (𝜑 → ∃𝑓 ∈ (Fil‘𝑌)(𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓)))

Distinct variable groups: 𝐵,𝑓 𝑓,𝐹 𝑓,𝐿 𝑓,𝑋 𝑓,𝑌 Allowed substitution hint: 𝜑(𝑓)

Proof of Theorem fmfnfm Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmfnfm.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌))
2		fbsspw 23894	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝒫 𝑌)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝑌)
4		elfvdm 6903	. . . . . . . 8 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ dom fBas)
6		fmfnfm.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋))
7		fmfnfm.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
8		fmfnfm.fm	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
9		ffn 6693	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
10		dffn4 6786	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑌 ↔ 𝐹:𝑌–onto→ran 𝐹)
11	9, 10	sylib 220	. . . . . . . . . 10 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌–onto→ran 𝐹)
12		foima 6785	. . . . . . . . . 10 ⊢ (𝐹:𝑌–onto→ran 𝐹 → (𝐹 “ 𝑌) = ran 𝐹)
13	7, 11, 12	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) = ran 𝐹)
14		filtop 23917	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
15	6, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐿)
16		fgcl 23940	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
17		filtop 23917	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
18	1, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐵))
19		eqid 2764	. . . . . . . . . . 11 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
20	19	imaelfm 24013	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
21	15, 1, 7, 18, 20	syl31anc 1394	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
22	13, 21	eqeltrrd 2865	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
23	8, 22	sseldd 3939	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ∈ 𝐿)
24		rnelfmlem 24014	. . . . . . 7 ⊢ (((𝑌 ∈ dom fBas ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
25	5, 6, 7, 23, 24	syl31anc 1394	. . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
26		fbsspw 23894	. . . . . 6 ⊢ (ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌)
27	25, 26	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ 𝒫 𝑌)
28	3, 27	unssd 4146	. . . 4 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ 𝒫 𝑌)
29		ssun1 4132	. . . . 5 ⊢ 𝐵 ⊆ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
30		fbasne0 23892	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ≠ ∅)
31	1, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ ∅)
32		ssn0 4360	. . . . 5 ⊢ ((𝐵 ⊆ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∧ 𝐵 ≠ ∅) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅)
33	29, 31, 32	sylancr 596	. . . 4 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅)
34		eqid 2764	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
35	34	elrnmpt 5936	. . . . . . . . 9 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥)))
36	35	elv 3461	. . . . . . . 8 ⊢ (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥))
37		0nelfil 23911	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐿)
38	6, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ∅ ∈ 𝐿)
39	38	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ¬ ∅ ∈ 𝐿)
40	6	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝐿 ∈ (Fil‘𝑋))
41	8	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
42	15, 1, 7	3jca 1142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
43	42	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
44		ssfg 23934	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
45	1, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen𝐵))
46	45	sselda 3938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ (𝑌filGen𝐵))
47	19	imaelfm 24013	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
48	43, 46, 47	syl2anc 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
49	41, 48	sseldd 3939	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ 𝐿)
50	40, 49	jca 519	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿))
51		filin 23916	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
52	51	3expa 1132	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
53	50, 52	sylan 589	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
54		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (((𝐹 “ 𝑠) ∩ 𝑥) = ∅ → (((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿 ↔ ∅ ∈ 𝐿))
55	53, 54	syl5ibcom 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (((𝐹 “ 𝑠) ∩ 𝑥) = ∅ → ∅ ∈ 𝐿))
56	39, 55	mtod 200	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ¬ ((𝐹 “ 𝑠) ∩ 𝑥) = ∅)
57		neq0 4306	. . . . . . . . . . . 12 ⊢ (¬ ((𝐹 “ 𝑠) ∩ 𝑥) = ∅ ↔ ∃𝑡 𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥))
58		elin 3922	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) ↔ (𝑡 ∈ (𝐹 “ 𝑠) ∧ 𝑡 ∈ 𝑥))
59		ffun 6696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
60		fvelima 6934	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐹 ∧ 𝑡 ∈ (𝐹 “ 𝑠)) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡)
61	60	ex 416	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun 𝐹 → (𝑡 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡))
62	7, 59, 61	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑡 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡))
63	62	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑡 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡))
64	7, 59	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun 𝐹)
65	64	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → Fun 𝐹)
66		fbelss 23895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
67	1, 66	sylan 589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
68	7	fdmd 6704	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom 𝐹 = 𝑌)
69	68	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → dom 𝐹 = 𝑌)
70	67, 69	sseqtrrd 3975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ dom 𝐹)
71	70	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → 𝑠 ⊆ dom 𝐹)
72	71	sselda 3938	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ dom 𝐹)
73		fvimacnv 7036	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
74	65, 72, 73	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
75		inelcm 4421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑠 ∧ 𝑦 ∈ (^◡𝐹 “ 𝑥)) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)
76	75	ex 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑠 → (𝑦 ∈ (^◡𝐹 “ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
77	76	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → (𝑦 ∈ (^◡𝐹 “ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
78	74, 77	sylbid 242	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
79		eleq1 2852	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = 𝑡 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑡 ∈ 𝑥))
80	79	imbi1d 343	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑦) = 𝑡 → (((𝐹‘𝑦) ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅) ↔ (𝑡 ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)))
81	78, 80	syl5ibcom 247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) = 𝑡 → (𝑡 ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)))
82	81	rexlimdva 3165	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡 → (𝑡 ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)))
83	63, 82	syld 47	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑡 ∈ (𝐹 “ 𝑠) → (𝑡 ∈ 𝑥 → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)))
84	83	impd 414	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ((𝑡 ∈ (𝐹 “ 𝑠) ∧ 𝑡 ∈ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
85	58, 84	biimtrid 244	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
86	85	exlimdv 1955	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (∃𝑡 𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
87	57, 86	biimtrid 244	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (¬ ((𝐹 “ 𝑠) ∩ 𝑥) = ∅ → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
88	56, 87	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)
89		ineq2 4168	. . . . . . . . . . 11 ⊢ (𝑡 = (^◡𝐹 “ 𝑥) → (𝑠 ∩ 𝑡) = (𝑠 ∩ (^◡𝐹 “ 𝑥)))
90	89	neeq1d 3018	. . . . . . . . . 10 ⊢ (𝑡 = (^◡𝐹 “ 𝑥) → ((𝑠 ∩ 𝑡) ≠ ∅ ↔ (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
91	88, 90	syl5ibrcom 249	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑡 = (^◡𝐹 “ 𝑥) → (𝑠 ∩ 𝑡) ≠ ∅))
92	91	rexlimdva 3165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥) → (𝑠 ∩ 𝑡) ≠ ∅))
93	36, 92	biimtrid 244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅))
94	93	expimpd 457	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ 𝐵 ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → (𝑠 ∩ 𝑡) ≠ ∅))
95	94	ralrimivv 3205	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ≠ ∅)
96		fbunfip 23931	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ↔ ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ≠ ∅))
97	1, 25, 96	syl2anc 593	. . . . 5 ⊢ (𝜑 → (¬ ∅ ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ↔ ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ≠ ∅))
98	95, 97	mpbird 259	. . . 4 ⊢ (𝜑 → ¬ ∅ ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
99		fsubbas 23929	. . . . 5 ⊢ (𝑌 ∈ dom fBas → ((fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) ↔ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ 𝒫 𝑌 ∧ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
100	1, 4, 99	3syl 18	. . . 4 ⊢ (𝜑 → ((fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) ↔ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ 𝒫 𝑌 ∧ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
101	28, 33, 98, 100	mpbir3and 1357	. . 3 ⊢ (𝜑 → (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌))
102		fgcl 23940	. . 3 ⊢ ((fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) → (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ∈ (Fil‘𝑌))
103	101, 102	syl 17	. 2 ⊢ (𝜑 → (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ∈ (Fil‘𝑌))
104		unexg 7728	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
105	1, 25, 104	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
106		ssfii 9367	. . . . 5 ⊢ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
107	105, 106	syl 17	. . . 4 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
108	107	unssad 4147	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
109		ssfg 23934	. . . 4 ⊢ ((fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) → (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
110	101, 109	syl 17	. . 3 ⊢ (𝜑 → (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
111	108, 110	sstrd 3948	. 2 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
112	1, 6, 7, 8	fmfnfmlem4 24019	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
113		elfm 24009	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
114	15, 101, 7, 113	syl3anc 1392	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
115	112, 114	bitr4d 284	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ 𝑡 ∈ ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
116	115	eqrdv 2762	. . 3 ⊢ (𝜑 → 𝐿 = ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
117		eqid 2764	. . . . 5 ⊢ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
118	117	fmfg 24011	. . . 4 ⊢ ((𝑋 ∈ 𝐿 ∧ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
119	15, 101, 7, 118	syl3anc 1392	. . 3 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
120	116, 119	eqtrd 2799	. 2 ⊢ (𝜑 → 𝐿 = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
121		sseq2 3964	. . . 4 ⊢ (𝑓 = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → (𝐵 ⊆ 𝑓 ↔ 𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
122		fveq2 6869	. . . . 5 ⊢ (𝑓 = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → ((𝑋 FilMap 𝐹)‘𝑓) = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
123	122	eqeq2d 2775	. . . 4 ⊢ (𝑓 = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → (𝐿 = ((𝑋 FilMap 𝐹)‘𝑓) ↔ 𝐿 = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))))
124	121, 123	anbi12d 641	. . 3 ⊢ (𝑓 = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → ((𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓)) ↔ (𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))))
125	124	rspcev 3583	. 2 ⊢ (((𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ∈ (Fil‘𝑌) ∧ (𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))) → ∃𝑓 ∈ (Fil‘𝑌)(𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓)))
126	103, 111, 120, 125	syl12anc 847	1 ⊢ (𝜑 → ∃𝑓 ∈ (Fil‘𝑌)(𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∃wex 1801 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 𝒫 cpw 4557 ↦ cmpt 5183 ^◡ccnv 5648 dom cdm 5649 ran crn 5650 “ cima 5652 Fun wfun 6517 Fn wfn 6518 ⟶wf 6519 –onto→wfo 6521 ‘cfv 6523 (class class class)co 7398 ficfi 9358 fBascfbas 21414 filGencfg 21415 Filcfil 23907 FilMap cfm 23995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1o 8439 df-2o 8440 df-en 8930 df-fin 8933 df-fi 9359 df-fbas 21423 df-fg 21424 df-fil 23908 df-fm 24000

This theorem is referenced by: fmufil 24021 cnpfcf 24103

W3C validator