Theorem rnelfm 23304

Description: A condition for a filter to be an image filter for a given function. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Assertion

Ref	Expression
rnelfm	⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹 ∈ 𝐿))

Proof of Theorem rnelfm

Dummy variables 𝑏 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		filtop 23206	. . . . . . 7 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
2	1	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐿)
3		simp1 1136	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ 𝐴)
4		simp3 1138	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
5		fmf 23296	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝑌 ∈ 𝐴 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
6	2, 3, 4, 5	syl3anc 1371	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
7	6	ffnd 6669	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8		fvelrnb 6903	. . . 4 ⊢ ((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿))
9	7, 8	syl 17	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿))
10		ffn 6668	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
11		dffn4 6762	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑌 ↔ 𝐹:𝑌–onto→ran 𝐹)
12	10, 11	sylib 217	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌–onto→ran 𝐹)
13		foima 6761	. . . . . . . . . . 11 ⊢ (𝐹:𝑌–onto→ran 𝐹 → (𝐹 “ 𝑌) = ran 𝐹)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝑌⟶𝑋 → (𝐹 “ 𝑌) = ran 𝐹)
15	14	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) = ran 𝐹)
16		simpll 765	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋 ∈ 𝐿)
17		simpr 485	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18		simplr 767	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
19		fgcl 23229	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20		filtop 23206	. . . . . . . . . . . 12 ⊢ ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
22	21	adantl 482	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑌filGen𝑏) = (𝑌filGen𝑏)
24	23	imaelfm 23302	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
25	16, 17, 18, 22, 24	syl31anc 1373	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
26	15, 25	eqeltrrd 2839	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27		eleq2 2826	. . . . . . . 8 ⊢ (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → (ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏) ↔ ran 𝐹 ∈ 𝐿))
28	26, 27	syl5ibcom 244	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿))
29	28	ex 413	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
30	1, 29	sylan 580	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
31	30	3adant1 1130	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
32	31	rexlimdv 3150	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿))
33	9, 32	sylbid 239	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) → ran 𝐹 ∈ 𝐿))
34		simpl2 1192	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑋))
35		filelss 23203	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
36	35	ex 413	. . . . . . . . 9 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋))
37	34, 36	syl 17	. . . . . . . 8 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋))
38		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → 𝑡 ∈ 𝐿)
39		eqidd 2737	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑡))
40		imaeq2 6009	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑡))
41	40	rspceeqv 3595	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐿 ∧ (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥))
42	38, 39, 41	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥))
43		simpl1 1191	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ 𝐴)
44		cnvimass 6033	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹 “ 𝑡) ⊆ dom 𝐹
45		fdm 6677	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
46	44, 45	sseqtrid 3996	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ 𝑡) ⊆ 𝑌)
47	46	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (◡𝐹 “ 𝑡) ⊆ 𝑌)
48	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑡) ⊆ 𝑌)
49	43, 48	ssexd 5281	. . . . . . . . . . . . 13 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑡) ∈ V)
50		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
51	50	elrnmpt 5911	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ 𝑡) ∈ V → ((◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥)))
52	49, 51	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥)))
53	52	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ((◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥)))
54	42, 53	mpbird 256	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
55		ssid 3966	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)
56		ffun 6671	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
57	56	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → Fun 𝐹)
58	57	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → Fun 𝐹)
59		funimass3 7004	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑡) ⊆ dom 𝐹) → ((𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡 ↔ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)))
60	58, 44, 59	sylancl 586	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ((𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡 ↔ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)))
61	55, 60	mpbiri 257	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡)
62		imaeq2 6009	. . . . . . . . . . . 12 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑡)))
63	62	sseq1d 3975	. . . . . . . . . . 11 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡))
64	63	rspcev 3581	. . . . . . . . . 10 ⊢ (((◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡) → ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)
65	54, 61, 64	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)
66	65	ex 413	. . . . . . . 8 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ 𝐿 → ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡))
67	37, 66	jcad 513	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ 𝐿 → (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)))
68	34	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ 𝑠) ⊆ 𝑡) ∧ 𝑡 ⊆ 𝑋)) → 𝐿 ∈ (Fil‘𝑋))
69	50	elrnmpt 5911	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
70	69	elv 3451	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
71		ssid 3966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)
72	57	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → Fun 𝐹)
73		cnvimass 6033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
74		funimass3 7004	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑥) ⊆ dom 𝐹) → ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥 ↔ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)))
75	72, 73, 74	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥 ↔ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)))
76	71, 75	mpbiri 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥)
77		imassrn 6024	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ ran 𝐹
78		ssin 4190	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥 ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ ran 𝐹) ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
79	76, 77, 78	sylanblc 589	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
80		elin 3926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹))
81		fvelrnb 6903	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹 Fn 𝑌 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
82	10, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:𝑌⟶𝑋 → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
83	82	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
84	83	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
85	72	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘𝑦) ∈ 𝑥) → Fun 𝐹)
86	85, 73	jctir 521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘𝑦) ∈ 𝑥) → (Fun 𝐹 ∧ (◡𝐹 “ 𝑥) ⊆ dom 𝐹))
87	57	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → Fun 𝐹)
88	87	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) → Fun 𝐹)
89	45	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
90	89	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → dom 𝐹 = 𝑌)
91	90	eleq2d 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝑌))
92	91	biimpar 478	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ dom 𝐹)
93		fvimacnv 7003	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (◡𝐹 “ 𝑥)))
94	88, 92, 93	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (◡𝐹 “ 𝑥)))
95	94	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘𝑦) ∈ 𝑥) → 𝑦 ∈ (◡𝐹 “ 𝑥))
96		funfvima2 7181	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑥) ⊆ dom 𝐹) → (𝑦 ∈ (◡𝐹 “ 𝑥) → (𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
97	86, 95, 96	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘𝑦) ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥)))
98	97	ex 413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
99		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
100		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥)) ↔ 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
101	99, 100	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑦) = 𝑧 → (((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥))) ↔ (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥)))))
102	98, 101	syl5ibcom 244	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑦) = 𝑧 → (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥)))))
103	102	rexlimdva 3152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧 → (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥)))))
104	84, 103	sylbid 239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑧 ∈ ran 𝐹 → (𝑧 ∈ 𝑥 → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥)))))
105	104	impcomd 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
106	80, 105	biimtrid 241	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑧 ∈ (𝑥 ∩ ran 𝐹) → 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
107	106	ssrdv 3950	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ (𝐹 “ (◡𝐹 “ 𝑥)))
108	79, 107	eqssd 3961	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (◡𝐹 “ 𝑥)) = (𝑥 ∩ ran 𝐹))
109		filin 23205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
110	109	3exp 1119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 → (ran 𝐹 ∈ 𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
111	110	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐿 ∈ (Fil‘𝑋) → (ran 𝐹 ∈ 𝐿 → (𝑥 ∈ 𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
112	111	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (ran 𝐹 ∈ 𝐿 → (𝑥 ∈ 𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113	112	imp31 418	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114	113	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
115	108, 114	eqeltrd 2838	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿)
116	115	exp32 421	. . . . . . . . . . . . . . 15 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿)))
117		imaeq2 6009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑥)))
118	117	sseq1d 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡))
119	117	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ∈ 𝐿 ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿))
120	119	imbi2d 340	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿) ↔ (𝑡 ⊆ 𝑋 → (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿)))
121	118, 120	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿)) ↔ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿))))
122	116, 121	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿))))
123	122	rexlimdva 3152	. . . . . . . . . . . . 13 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿))))
124	70, 123	biimtrid 241	. . . . . . . . . . . 12 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿))))
125	124	imp44 429	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ 𝑠) ⊆ 𝑡) ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ 𝑠) ∈ 𝐿)
126		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ 𝑠) ⊆ 𝑡) ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ⊆ 𝑋)
127		simprlr 778	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ 𝑠) ⊆ 𝑡) ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ 𝑠) ⊆ 𝑡)
128		filss 23204	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ (𝐹 “ 𝑠) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
129	68, 125, 126, 127, 128	syl13anc 1372	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ (𝐹 “ 𝑠) ⊆ 𝑡) ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ∈ 𝐿)
130	129	exp44 438	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
131	130	rexlimdv 3150	. . . . . . . 8 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
132	131	impcomd 412	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡) → 𝑡 ∈ 𝐿))
133	67, 132	impbid 211	. . . . . 6 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)))
134	2	adantr 481	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑋 ∈ 𝐿)
135		rnelfmlem 23303	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
136		simpl3 1193	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
137		elfm 23298	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐿 ∧ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)))
138	134, 135, 136, 137	syl3anc 1371	. . . . . 6 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝐹 “ 𝑠) ⊆ 𝑡)))
139	133, 138	bitr4d 281	. . . . 5 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑡 ∈ 𝐿 ↔ 𝑡 ∈ ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))))
140	139	eqrdv 2734	. . . 4 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
141	7	adantr 481	. . . . 5 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142		fnfvelrn 7031	. . . . 5 ⊢ (((𝑋 FilMap 𝐹) Fn (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) → ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ∈ ran (𝑋 FilMap 𝐹))
143	141, 135, 142	syl2anc 584	. . . 4 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ∈ ran (𝑋 FilMap 𝐹))
144	140, 143	eqeltrd 2838	. . 3 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145	144	ex 413	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (ran 𝐹 ∈ 𝐿 → 𝐿 ∈ ran (𝑋 FilMap 𝐹)))
146	33, 145	impbid 211	1 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹 ∈ 𝐿))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910   ↦ cmpt 5188  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7357  fBascfbas 20784  filGencfg 20785  Filcfil 23196   FilMap cfm 23284

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-fbas 20793  df-fg 20794  df-fil 23197  df-fm 23289

This theorem is referenced by: (None)