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Theorem fmfnfmlem4 22269

Description: Lemma for fmfnfm 22270. (Contributed by Jeff Hankins, 19-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
fmfnfmlem4	⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))

Distinct variable groups: 𝑡,𝑠,𝑥,𝐵 𝐹,𝑠,𝑡,𝑥 𝐿,𝑠,𝑡,𝑥 𝜑,𝑠,𝑡,𝑥 𝑋,𝑠,𝑡,𝑥 𝑌,𝑠,𝑡,𝑥

Proof of Theorem fmfnfmlem4 Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmfnfm.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋))
2		filelss 22164	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
3	2	ex 405	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋))
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋))
5		fmfnfm.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌))
6		mptexg 6810	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
7		rnexg 7429	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝐿 ∈ (Fil‘𝑋) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
9	1, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
10		unexg 7289	. . . . . . . . 9 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
11	5, 9, 10	syl2anc 576	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
12		ssfii 8678	. . . . . . . . 9 ⊢ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
13	12	unssbd 4053	. . . . . . . 8 ⊢ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
14	11, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
15	14	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
16		eqid 2779	. . . . . . . . 9 ⊢ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)
17		imaeq2 5766	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝑡))
18	17	rspceeqv 3554	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝐿 ∧ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
19	16, 18	mpan2 678	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
20	19	adantl 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
21		elfvdm 6531	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
22	5, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ dom fBas)
23		cnvimass 5789	. . . . . . . . . . 11 ⊢ (^◡𝐹 “ 𝑡) ⊆ dom 𝐹
24		fmfnfm.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
25	23, 24	fssdm 6360	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
26	22, 25	ssexd 5084	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ∈ V)
27	26	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ V)
28		eqid 2779	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
29	28	elrnmpt 5671	. . . . . . . 8 ⊢ ((^◡𝐹 “ 𝑡) ∈ V → ((^◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥)))
30	27, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ((^◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥)))
31	20, 30	mpbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
32	15, 31	sseldd 3860	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
33		ffun 6347	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
34		ssid 3880	. . . . . . . 8 ⊢ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)
35		funimass2 6270	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)) → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
36	33, 34, 35	sylancl 577	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
37	24, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
38	37	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
39		imaeq2 5766	. . . . . . 7 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (^◡𝐹 “ 𝑡)))
40	39	sseq1d 3889	. . . . . 6 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡))
41	40	rspcev 3536	. . . . 5 ⊢ (((^◡𝐹 “ 𝑡) ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ∧ (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡) → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)
42	32, 38, 41	syl2anc 576	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)
43	42	ex 405	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡))
44	4, 43	jcad 505	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
45		elfiun 8689	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V) → (𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ↔ (𝑠 ∈ (fi‘𝐵) ∨ 𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∨ ∃𝑧 ∈ (fi‘𝐵)∃𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))𝑠 = (𝑧 ∩ 𝑤))))
46	5, 9, 45	syl2anc 576	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) ↔ (𝑠 ∈ (fi‘𝐵) ∨ 𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∨ ∃𝑧 ∈ (fi‘𝐵)∃𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))𝑠 = (𝑧 ∩ 𝑤))))
47		fmfnfm.fm	. . . . . . 7 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
48	5, 1, 24, 47	fmfnfmlem1 22266	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
49	5, 1, 24, 47	fmfnfmlem3 22268	. . . . . . . 8 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
50	49	eleq2d 2852	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
51	28	elrnmpt 5671	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥)))
52	51	elv 3421	. . . . . . . 8 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥))
53	5, 1, 24, 47	fmfnfmlem2 22267	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
54	52, 53	syl5bi 234	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
55	50, 54	sylbid 232	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
56	49	eleq2d 2852	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
57	28	elrnmpt 5671	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
58	57	elv 3421	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥))
59	56, 58	syl6bb 279	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
60	59	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
61		fbssfi 22149	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑧 ∈ (fi‘𝐵)) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧)
62	5, 61	sylan 572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧)
63	1	ad3antrrr 717	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝐿 ∈ (Fil‘𝑋))
64	1	adantr 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝐿 ∈ (Fil‘𝑋))
65	47	adantr 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
66		filtop 22167	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
67	1, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐿)
68	67, 5, 24	3jca 1108	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
69	68	adantr 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
70		ssfg 22184	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
71	5, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen𝐵))
72	71	sselda 3859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ (𝑌filGen𝐵))
73		eqid 2779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
74	73	imaelfm 22263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
75	69, 72, 74	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
76	65, 75	sseldd 3860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ 𝐿)
77	76	adantrr 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐹 “ 𝑠) ∈ 𝐿)
78	64, 77	jca 504	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿))
79		filin 22166	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
80	79	3expa 1098	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
81	78, 80	sylan 572	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
82	81	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
83		simprr 760	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ⊆ 𝑋)
84		elin 4058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) ↔ (𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥))
85	24, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → Fun 𝐹)
86		fvelima 6561	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ (𝐹 “ 𝑠)) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤)
87	86	ex 405	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (Fun 𝐹 → (𝑤 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤))
88	85, 87	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑤 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤))
89	88	ad3antrrr 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤))
90	85	ad3antrrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → Fun 𝐹)
91		simplrr 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → 𝑠 ⊆ 𝑧)
92		simprl 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ 𝑠)
93		ssel2 3854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑠 ⊆ 𝑧 ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ 𝑧)
94	91, 92, 93	syl2an 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ 𝑧)
95	85	ad2antrr 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → Fun 𝐹)
96		fbelss 22145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
97	5, 96	sylan 572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
98	24	fdmd 6353	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → dom 𝐹 = 𝑌)
99	98	adantr 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → dom 𝐹 = 𝑌)
100	97, 99	sseqtr4d 3899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ dom 𝐹)
101	100	adantrr 704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝑠 ⊆ dom 𝐹)
102	101	sselda 3859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ dom 𝐹)
103		fvimacnv 6648	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
104	95, 102, 103	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
105	104	biimpd 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → 𝑦 ∈ (^◡𝐹 “ 𝑥)))
106	105	impr 447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
107	106	ad2ant2rl 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
108	94, 107	elind 4060	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)))
109		inss2 4094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ (^◡𝐹 “ 𝑥)
110		cnvimass 5789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
111	109, 110	sstri 3868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹
112		funfvima2 6819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun 𝐹 ∧ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹) → (𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
113	111, 112	mpan2 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun 𝐹 → (𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
114	90, 108, 113	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
115	114	anassrs 460	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
116	115	expr 449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
117		eleq1 2854	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
118		eleq1 2854	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ↔ 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
119	117, 118	imbi12d 337	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑤 → (((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))) ↔ (𝑤 ∈ 𝑥 → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))))
120	116, 119	syl5ibcom 237	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) = 𝑤 → (𝑤 ∈ 𝑥 → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))))
121	120	rexlimdva 3230	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤 → (𝑤 ∈ 𝑥 → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))))
122	89, 121	syld 47	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ (𝐹 “ 𝑠) → (𝑤 ∈ 𝑥 → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))))
123	122	impd 402	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → ((𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
124	84, 123	syl5bi 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
125	124	adantrl 703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
126	125	ssrdv 3865	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
127		simprl 758	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡)
128	126, 127	sstrd 3869	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)
129		filss 22165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
130	63, 82, 83, 128, 129	syl13anc 1352	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ∈ 𝐿)
131	130	exp32 413	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
132		ineq2 4071	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝑧 ∩ 𝑤) = (𝑧 ∩ (^◡𝐹 “ 𝑥)))
133	132	imaeq2d 5770	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝐹 “ (𝑧 ∩ 𝑤)) = (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
134	133	sseq1d 3889	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 ↔ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡))
135	134	imbi1d 334	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)) ↔ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
136	131, 135	syl5ibrcom 239	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → (𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
137	136	rexlimdva 3230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
138	137	rexlimdvaa 3231	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧 → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))))
139	138	imp 398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
140	62, 139	syldan 582	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
141	60, 140	sylbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
142	141	impr 447	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (fi‘𝐵) ∧ 𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
143		imaeq2 5766	. . . . . . . . . 10 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → (𝐹 “ 𝑠) = (𝐹 “ (𝑧 ∩ 𝑤)))
144	143	sseq1d 3889	. . . . . . . . 9 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡))
145	144	imbi1d 334	. . . . . . . 8 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → (((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)) ↔ ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
146	142, 145	syl5ibrcom 239	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (fi‘𝐵) ∧ 𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → (𝑠 = (𝑧 ∩ 𝑤) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
147	146	rexlimdvva 3240	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ (fi‘𝐵)∃𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))𝑠 = (𝑧 ∩ 𝑤) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
148	48, 55, 147	3jaod 1408	. . . . 5 ⊢ (𝜑 → ((𝑠 ∈ (fi‘𝐵) ∨ 𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∨ ∃𝑧 ∈ (fi‘𝐵)∃𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))𝑠 = (𝑧 ∩ 𝑤)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
149	46, 148	sylbid 232	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
150	149	rexlimdv 3229	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
151	150	impcomd 403	. 2 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡) → 𝑡 ∈ 𝐿))
152	44, 151	impbid 204	1 ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ w3o 1067 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∃wrex 3090 Vcvv 3416 ∪ cun 3828 ∩ cin 3829 ⊆ wss 3830 ↦ cmpt 5008 ^◡ccnv 5406 dom cdm 5407 ran crn 5408 “ cima 5410 Fun wfun 6182 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ficfi 8669 fBascfbas 20235 filGencfg 20236 Filcfil 22157 FilMap cfm 22245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-fin 8310 df-fi 8670 df-fbas 20244 df-fg 20245 df-fil 22158 df-fm 22250

This theorem is referenced by: fmfnfm 22270

W3C validator