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Theorem fmfnfm 22560

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 22434	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ 𝒫 𝑌)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝑌)
4		elfvdm 6696	. . . . . . . 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 6508	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
10		dffn4 6590	. . . . . . . . . . 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 6589	. . . . . . . . . 10 ⊢ (𝐹:𝑌–onto→ran 𝐹 → (𝐹 “ 𝑌) = ran 𝐹)
13	7, 11, 12	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) = ran 𝐹)
14		filtop 22457	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
15	6, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐿)
16		fgcl 22480	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
17		filtop 22457	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
18	1, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐵))
19		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
20	19	imaelfm 22553	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
21	15, 1, 7, 18, 20	syl31anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
22	13, 21	eqeltrrd 2914	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
23	8, 22	sseldd 3967	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ∈ 𝐿)
24		rnelfmlem 22554	. . . . . . 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 1369	. . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
26		fbsspw 22434	. . . . . 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 4161	. . . 4 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ 𝒫 𝑌)
29		ssun1 4147	. . . . 5 ⊢ 𝐵 ⊆ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
30		fbasne0 22432	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ≠ ∅)
31	1, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ ∅)
32		ssn0 4353	. . . . 5 ⊢ ((𝐵 ⊆ (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∧ 𝐵 ≠ ∅) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅)
33	29, 31, 32	sylancr 589	. . . 4 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ≠ ∅)
34		eqid 2821	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
35	34	elrnmpt 5822	. . . . . . . . 9 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥)))
36	35	elv 3499	. . . . . . . 8 ⊢ (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥))
37		0nelfil 22451	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐿)
38	6, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ ∅ ∈ 𝐿)
39	38	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ¬ ∅ ∈ 𝐿)
40	6	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝐿 ∈ (Fil‘𝑋))
41	8	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
42	15, 1, 7	3jca 1124	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
43	42	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
44		ssfg 22474	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
45	1, 44	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen𝐵))
46	45	sselda 3966	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ (𝑌filGen𝐵))
47	19	imaelfm 22553	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
48	43, 46, 47	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
49	41, 48	sseldd 3967	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ 𝐿)
50	40, 49	jca 514	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿))
51		filin 22456	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
52	51	3expa 1114	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
53	50, 52	sylan 582	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
54		eleq1 2900	. . . . . . . . . . . . 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 4308	. . . . . . . . . . . 12 ⊢ (¬ ((𝐹 “ 𝑠) ∩ 𝑥) = ∅ ↔ ∃𝑡 𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥))
58		elin 4168	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) ↔ (𝑡 ∈ (𝐹 “ 𝑠) ∧ 𝑡 ∈ 𝑥))
59		ffun 6511	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
60		fvelima 6725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝐹 ∧ 𝑡 ∈ (𝐹 “ 𝑠)) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑡)
61	60	ex 415	. . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → Fun 𝐹)
66		fbelss 22435	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
67	1, 66	sylan 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
68	7	fdmd 6517	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom 𝐹 = 𝑌)
69	68	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → dom 𝐹 = 𝑌)
70	67, 69	sseqtrrd 4007	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ dom 𝐹)
71	70	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → 𝑠 ⊆ dom 𝐹)
72	71	sselda 3966	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ dom 𝐹)
73		fvimacnv 6817	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
74	65, 72, 73	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
75		inelcm 4413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑠 ∧ 𝑦 ∈ (^◡𝐹 “ 𝑥)) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅)
76	75	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑠 → (𝑦 ∈ (^◡𝐹 “ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
77	76	adantl 484	. . . . . . . . . . . . . . . . . . 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 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = 𝑡 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑡 ∈ 𝑥))
80	79	imbi1d 344	. . . . . . . . . . . . . . . . . 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 3284	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → ((𝑡 ∈ (𝐹 “ 𝑠) ∧ 𝑡 ∈ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
85	58, 84	syl5bi 244	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
86	85	exlimdv 1930	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐵) ∧ 𝑥 ∈ 𝐿) → (∃𝑡 𝑡 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → (𝑠 ∩ (^◡𝐹 “ 𝑥)) ≠ ∅))
87	57, 86	syl5bi 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 4182	. . . . . . . . . . 11 ⊢ (𝑡 = (^◡𝐹 “ 𝑥) → (𝑠 ∩ 𝑡) = (𝑠 ∩ (^◡𝐹 “ 𝑥)))
90	89	neeq1d 3075	. . . . . . . . . 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 3284	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (∃𝑥 ∈ 𝐿 𝑡 = (^◡𝐹 “ 𝑥) → (𝑠 ∩ 𝑡) ≠ ∅))
93	36, 92	syl5bi 244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) → (𝑠 ∩ 𝑡) ≠ ∅))
94	93	expimpd 456	. . . . . 6 ⊢ (𝜑 → ((𝑠 ∈ 𝐵 ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → (𝑠 ∩ 𝑡) ≠ ∅))
95	94	ralrimivv 3190	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝐵 ∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ≠ ∅)
96		fbunfip 22471	. . . . . 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 586	. . . . 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 22469	. . . . 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 1338	. . 3 ⊢ (𝜑 → (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∈ (fBas‘𝑌))
102		fgcl 22480	. . 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 7466	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
105	1, 25, 104	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
106		ssfii 8877	. . . . 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 4162	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
109		ssfg 22474	. . . 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 3976	. 2 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
112	1, 6, 7, 8	fmfnfmlem4 22559	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
113		elfm 22549	. . . . . 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 1367	. . . . 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 2819	. . 3 ⊢ (𝜑 → 𝐿 = ((𝑋 FilMap 𝐹)‘(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))))
117		eqid 2821	. . . . 5 ⊢ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
118	117	fmfg 22551	. . . 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 1367	. . 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 2856	. 2 ⊢ (𝜑 → 𝐿 = ((𝑋 FilMap 𝐹)‘(𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
121		sseq2 3992	. . . 4 ⊢ (𝑓 = (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → (𝐵 ⊆ 𝑓 ↔ 𝐵 ⊆ (𝑌filGen(fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))))
122		fveq2 6664	. . . . 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 2832	. . . 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 632	. . 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 3622	. 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 834	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 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ↦ cmpt 5138 ^◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 ficfi 8868 fBascfbas 20527 filGencfg 20528 Filcfil 22447 FilMap cfm 22535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-fbas 20536 df-fg 20537 df-fil 22448 df-fm 22540

This theorem is referenced by: fmufil 22561 cnpfcf 22643

W3C validator