Theorem rnelfm 23104

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 23006	. . . . . . 7 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
2	1	3ad2ant2 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐿)
3		simp1 1135	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ 𝐴)
4		simp3 1137	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
5		fmf 23096	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝑌 ∈ 𝐴 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
6	2, 3, 4, 5	syl3anc 1370	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
7	6	ffnd 6601	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8		fvelrnb 6830	. . . 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 6600	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
11		dffn4 6694	. . . . . . . . . . . 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 6693	. . . . . . . . . . 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 724	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) = ran 𝐹)
16		simpll 764	. . . . . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
19		fgcl 23029	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20		filtop 23006	. . . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ (𝑌filGen𝑏) = (𝑌filGen𝑏)
24	23	imaelfm 23102	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
25	16, 17, 18, 22, 24	syl31anc 1372	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
26	15, 25	eqeltrrd 2840	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27		eleq2 2827	. . . . . . . 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 1129	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
32	31	rexlimdv 3212	. . 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 1191	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑋))
35		filelss 23003	. . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑡))
40		imaeq2 5965	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑡))
41	40	rspceeqv 3575	. . . . . . . . . . . 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 1190	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ 𝐴)
44		cnvimass 5989	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹 “ 𝑡) ⊆ dom 𝐹
45		fdm 6609	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
46	44, 45	sseqtrid 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ 𝑡) ⊆ 𝑌)
47	46	3ad2ant3 1134	. . . . . . . . . . . . . . 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 5248	. . . . . . . . . . . . 13 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑡) ∈ V)
50		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
51	50	elrnmpt 5865	. . . . . . . . . . . . 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 3943	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)
56		ffun 6603	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
57	56	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → Fun 𝐹)
58	57	ad2antrr 723	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → Fun 𝐹)
59		funimass3 6931	. . . . . . . . . . . 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 5965	. . . . . . . . . . . 12 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑡)))
63	62	sseq1d 3952	. . . . . . . . . . 11 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡))
64	63	rspcev 3561	. . . . . . . . . 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 5865	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
70	69	elv 3438	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
71		ssid 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)
72	57	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → Fun 𝐹)
73		cnvimass 5989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
74		funimass3 6931	. . . . . . . . . . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ ran 𝐹
78		ssin 4164	. . . . . . . . . . . . . . . . . . 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 3903	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹))
81		fvelrnb 6830	. . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
84	83	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → Fun 𝐹)
88	87	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
90	89	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7107	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
100		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹 “ (◡𝐹 “ 𝑥)) ↔ 𝑧 ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
101	99, 100	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . . . . . . . . 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	syl5bi 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 3927	. . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . 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 23005	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
110	109	3exp 1118	. . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 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 5965	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑥)))
118	117	sseq1d 3952	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡))
119	117	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ∈ 𝐿 ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿))
120	119	imbi2d 341	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿) ↔ (𝑡 ⊆ 𝑋 → (𝐹 “ (◡𝐹 “ 𝑥)) ∈ 𝐿)))
121	118, 120	imbi12d 345	. . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . 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	syl5bi 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 770	. . . . . . . . . . 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 777	. . . . . . . . . . 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 23004	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ (𝐹 “ 𝑠) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
129	68, 125, 126, 127, 128	syl13anc 1371	. . . . . . . . . 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 3212	. . . . . . . 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 23103	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
136		simpl3 1192	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
137		elfm 23098	. . . . . . 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 1370	. . . . . 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 2736	. . . 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 6958	. . . . 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 2839	. . 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 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275  fBascfbas 20585  filGencfg 20586  Filcfil 22996   FilMap cfm 23084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-fbas 20594  df-fg 20595  df-fil 22997  df-fm 23089

This theorem is referenced by: (None)