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Theorem rnelfm 23677

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 23579	. . . . . . 7 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
2	1	3ad2ant2 1132	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ 𝐿)
3		simp1 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ 𝐴)
4		simp3 1136	. . . . . 6 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
5		fmf 23669	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝑌 ∈ 𝐴 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
6	2, 3, 4, 5	syl3anc 1369	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))
7	6	ffnd 6717	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
8		fvelrnb 6951	. . . 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 6716	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
11		dffn4 6810	. . . . . . . . . . . 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 6809	. . . . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) = ran 𝐹)
16		simpll 763	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑋 ∈ 𝐿)
17		simpr 483	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑏 ∈ (fBas‘𝑌))
18		simplr 765	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝐹:𝑌⟶𝑋)
19		fgcl 23602	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (fBas‘𝑌) → (𝑌filGen𝑏) ∈ (Fil‘𝑌))
20		filtop 23579	. . . . . . . . . . . 12 ⊢ ((𝑌filGen𝑏) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (fBas‘𝑌) → 𝑌 ∈ (𝑌filGen𝑏))
22	21	adantl 480	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → 𝑌 ∈ (𝑌filGen𝑏))
23		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑌filGen𝑏) = (𝑌filGen𝑏)
24	23	imaelfm 23675	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝑏 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝑏)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
25	16, 17, 18, 22, 24	syl31anc 1371	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝑏))
26	15, 25	eqeltrrd 2832	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑏 ∈ (fBas‘𝑌)) → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝑏))
27		eleq2 2820	. . . . . . . 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 411	. . . . . 6 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
30	1, 29	sylan 578	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
31	30	3adant1 1128	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) → (((𝑋 FilMap 𝐹)‘𝑏) = 𝐿 → ran 𝐹 ∈ 𝐿)))
32	31	rexlimdv 3151	. . 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 1190	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑋))
35		filelss 23576	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
36	35	ex 411	. . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → 𝑡 ∈ 𝐿)
39		eqidd 2731	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡))
40		imaeq2 6054	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝑡))
41	40	rspceeqv 3632	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐿 ∧ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
42	38, 39, 41	syl2anc 582	. . . . . . . . . . 11 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
43		simpl1 1189	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑌 ∈ 𝐴)
44		cnvimass 6079	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝐹 “ 𝑡) ⊆ dom 𝐹
45		fdm 6725	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
46	44, 45	sseqtrid 4033	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑌⟶𝑋 → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
47	46	3ad2ant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
48	47	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
49	43, 48	ssexd 5323	. . . . . . . . . . . . 13 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ V)
50		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
51	50	elrnmpt 5954	. . . . . . . . . . . . 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 479	. . . . . . . . . . 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 4003	. . . . . . . . . . 11 ⊢ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)
56		ffun 6719	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
57	56	3ad2ant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → Fun 𝐹)
58	57	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑡 ∈ 𝐿) → Fun 𝐹)
59		funimass3 7054	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ (^◡𝐹 “ 𝑡) ⊆ dom 𝐹) → ((𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡 ↔ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)))
60	58, 44, 59	sylancl 584	. . . . . . . . . . 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 6054	. . . . . . . . . . . 12 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (^◡𝐹 “ 𝑡)))
63	62	sseq1d 4012	. . . . . . . . . . 11 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡))
64	63	rspcev 3611	. . . . . . . . . 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 582	. . . . . . . . 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 411	. . . . . . . 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 511	. . . . . . 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 479	. . . . . . . . . . 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 5954	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥)))
70	69	elv 3478	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥))
71		ssid 4003	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝐹 “ 𝑥) ⊆ (^◡𝐹 “ 𝑥)
72	57	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → Fun 𝐹)
73		cnvimass 6079	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
74		funimass3 7054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝐹 ∧ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹) → ((𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ 𝑥 ↔ (^◡𝐹 “ 𝑥) ⊆ (^◡𝐹 “ 𝑥)))
75	72, 73, 74	sylancl 584	. . . . . . . . . . . . . . . . . . . 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 6069	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ ran 𝐹
78		ssin 4229	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ 𝑥 ∧ (𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ ran 𝐹) ↔ (𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ (𝑥 ∩ ran 𝐹))
79	76, 77, 78	sylanblc 587	. . . . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹))
81		fvelrnb 6951	. . . . . . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑌 (𝐹‘𝑦) = 𝑧))
84	83	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → Fun 𝐹)
88	87	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → dom 𝐹 = 𝑌)
90	89	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
94	88, 92, 93	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7234	. . . . . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
100		eleq1 2819	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹 “ (^◡𝐹 “ 𝑥)) ↔ 𝑧 ∈ (𝐹 “ (^◡𝐹 “ 𝑥))))
101	99, 100	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . . . . . 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 3998	. . . . . . . . . . . . . . . . 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 23578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
110	109	3exp 1117	. . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (ran 𝐹 ∈ 𝐿 → (𝑥 ∈ 𝐿 → (𝑥 ∩ ran 𝐹) ∈ 𝐿)))
113	112	imp31 416	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
114	113	adantr 479	. . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . 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 6054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (^◡𝐹 “ 𝑥) → (𝐹 “ 𝑠) = (𝐹 “ (^◡𝐹 “ 𝑥)))
118	117	sseq1d 4012	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (^◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (^◡𝐹 “ 𝑥)) ⊆ 𝑡))
119	117	eleq1d 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (^◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ∈ 𝐿 ↔ (𝐹 “ (^◡𝐹 “ 𝑥)) ∈ 𝐿))
120	119	imbi2d 339	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (^◡𝐹 “ 𝑥) → ((𝑡 ⊆ 𝑋 → (𝐹 “ 𝑠) ∈ 𝐿) ↔ (𝑡 ⊆ 𝑋 → (𝐹 “ (^◡𝐹 “ 𝑥)) ∈ 𝐿)))
121	118, 120	imbi12d 343	. . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . 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 427	. . . . . . . . . . 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 769	. . . . . . . . . . 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 776	. . . . . . . . . . 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 23577	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ (𝐹 “ 𝑠) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
129	68, 125, 126, 127, 128	syl13anc 1370	. . . . . . . . . 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 436	. . . . . . . . 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 3151	. . . . . . . 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 410	. . . . . . 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 479	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝑋 ∈ 𝐿)
135		rnelfmlem 23676	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌))
136		simpl3 1191	. . . . . . 7 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
137		elfm 23671	. . . . . . 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 1369	. . . . . 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 2728	. . . 4 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 = ((𝑋 FilMap 𝐹)‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
141	7	adantr 479	. . . . 5 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → (𝑋 FilMap 𝐹) Fn (fBas‘𝑌))
142		fnfvelrn 7081	. . . . 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 582	. . . 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 2831	. . 3 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → 𝐿 ∈ ran (𝑋 FilMap 𝐹))
145	144	ex 411	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∃wrex 3068 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 ↦ cmpt 5230 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 “ cima 5678 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7411 fBascfbas 21132 filGencfg 21133 Filcfil 23569 FilMap cfm 23657

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-fbas 21141 df-fg 21142 df-fil 23570 df-fm 23662

This theorem is referenced by: (None)

W3C validator