Theorem rnelfm 23901

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 23803	. . . . . . 7 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
2	1	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐿)
3		simp1 1137	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ 𝐴)
4		simp3 1139	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
5		fmf 23893	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝑌 ∈ 𝐴 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
6	2, 3, 4, 5	syl3anc 1374	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
7	6	ffnd 6664	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8		fvelrnb 6895	. . . 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 6663	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
11		dffn4 6753	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑌 ↔ 𝐹:𝑌–onto→ran 𝐹)
12	10, 11	sylib 218	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌–onto→ran 𝐹)
13		foima 6752	. . . . . . . . . . 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 728	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) = ran 𝐹)
16		simpll 767	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋 ∈ 𝐿)
17		simpr 484	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18		simplr 769	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
19		fgcl 23826	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20		filtop 23803	. . . . . . . . . . . 12 ⊢ ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
22	21	adantl 481	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑌filGen𝑏) = (𝑌filGen𝑏)
24	23	imaelfm 23899	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
25	16, 17, 18, 22, 24	syl31anc 1376	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
26	15, 25	eqeltrrd 2838	. . . . . . . 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 245	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿))
29	28	ex 412	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
30	1, 29	sylan 581	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
31	30	3adant1 1131	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
32	31	rexlimdv 3136	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑏 ∈ (fBas‘𝑌)((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿))
33	9, 32	sylbid 240	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) → ran 𝐹 ∈ 𝐿))
34		simpl2 1194	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑋))
35		filelss 23800	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
36	35	ex 412	. . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → 𝑡 ∈ 𝐿)
39		eqidd 2738	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑡))
40		imaeq2 6016	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑡))
41	40	rspceeqv 3600	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐿 ∧ (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥))
42	38, 39, 41	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (◡𝐹 “ 𝑡) = (◡𝐹 “ 𝑥))
43		simpl1 1193	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ 𝐴)
44		cnvimass 6042	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹 “ 𝑡) ⊆ dom 𝐹
45		fdm 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
46	44, 45	sseqtrid 3977	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑌⟶𝑋 → (◡𝐹 “ 𝑡) ⊆ 𝑌)
47	46	3ad2ant3 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (◡𝐹 “ 𝑡) ⊆ 𝑌)
48	47	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑡) ⊆ 𝑌)
49	43, 48	ssexd 5270	. . . . . . . . . . . . 13 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (◡𝐹 “ 𝑡) ∈ V)
50		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
51	50	elrnmpt 5908	. . . . . . . . . . . . 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 480	. . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
55		ssid 3957	. . . . . . . . . . 11 ⊢ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)
56		ffun 6666	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
57	56	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → Fun 𝐹)
58	57	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → Fun 𝐹)
59		funimass3 7001	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑡) ⊆ dom 𝐹) → ((𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡 ↔ (◡𝐹 “ 𝑡) ⊆ (◡𝐹 “ 𝑡)))
60	58, 44, 59	sylancl 587	. . . . . . . . . . 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 258	. . . . . . . . . 10 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡)
62		imaeq2 6016	. . . . . . . . . . . 12 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑡)))
63	62	sseq1d 3966	. . . . . . . . . . 11 ⊢ (𝑠 = (◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑡)) ⊆ 𝑡))
64	63	rspcev 3577	. . . . . . . . . 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 585	. . . . . . . . 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 412	. . . . . . . 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 512	. . . . . . 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 480	. . . . . . . . . . 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 5908	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
70	69	elv 3446	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
71		ssid 3957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)
72	57	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → Fun 𝐹)
73		cnvimass 6042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
74		funimass3 7001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑥) ⊆ dom 𝐹) → ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥 ↔ (◡𝐹 “ 𝑥) ⊆ (◡𝐹 “ 𝑥)))
75	72, 73, 74	sylancl 587	. . . . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . . . 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 6031	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ ran 𝐹
78		ssin 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑥 ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ ran 𝐹) ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
79	76, 77, 78	sylanblc 590	. . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹))
81		fvelrnb 6895	. . . . . . . . . . . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
84	83	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → Fun 𝐹)
88	87	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
90	89	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7000	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (◡𝐹 “ 𝑥)))
94	88, 92, 93	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7179	. . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . . . . . . 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 3138	. . . . . . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . . . . . 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 3952	. . . . . . . . . . . . . . . . 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 23802	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
110	109	3exp 1120	. . . . . . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (ran 𝐹 ∈ 𝐿 → (𝑥 ∈ 𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113	112	imp31 417	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114	113	adantr 480	. . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . 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 6016	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑥)))
118	117	sseq1d 3966	. . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . 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 3138	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 428	. . . . . . . . . . 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 773	. . . . . . . . . . 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 780	. . . . . . . . . . 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 23801	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ (𝐹 “ 𝑠) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
129	68, 125, 126, 127, 128	syl13anc 1375	. . . . . . . . . 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 437	. . . . . . . . 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 3136	. . . . . . . 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 411	. . . . . . 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 212	. . . . . 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 480	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑋 ∈ 𝐿)
135		rnelfmlem 23900	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
136		simpl3 1195	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
137		elfm 23895	. . . . . . 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 1374	. . . . . 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 282	. . . . 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 2735	. . . 4 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
141	7	adantr 480	. . . . 5 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142		fnfvelrn 7027	. . . . 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 585	. . . 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 2837	. . 3 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145	144	ex 412	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (ran 𝐹 ∈ 𝐿 → 𝐿 ∈ ran (𝑋 FilMap 𝐹)))
146	33, 145	impbid 212	1 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹 ∈ 𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  –onto→wfo 6491  ‘cfv 6493  (class class class)co 7360  fBascfbas 21301  filGencfg 21302  Filcfil 23793   FilMap cfm 23881

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-fbas 21310  df-fg 21311  df-fil 23794  df-fm 23886

This theorem is referenced by: (None)