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Theorem fmfnfmlem4 23877

Description: Lemma for fmfnfm 23878. (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 23772	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
3	2	ex 412	. . . 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 7177	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
7		rnexg 7858	. . . . . . . . . . 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 7699	. . . . . . . . 9 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V) → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
11	5, 9, 10	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V)
12		ssfii 9346	. . . . . . . . 9 ⊢ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
13	12	unssbd 4153	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
16		eqid 2729	. . . . . . . . 9 ⊢ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)
17		imaeq2 6016	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝑡))
18	17	rspceeqv 3608	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝐿 ∧ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
19	16, 18	mpan2 691	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
20	19	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
21		elfvdm 6877	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
22	5, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ dom fBas)
23		cnvimass 6042	. . . . . . . . . . 11 ⊢ (^◡𝐹 “ 𝑡) ⊆ dom 𝐹
24		fmfnfm.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
25	23, 24	fssdm 6689	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
26	22, 25	ssexd 5274	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ∈ V)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ V)
28		eqid 2729	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
29	28	elrnmpt 5911	. . . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
32	15, 31	sseldd 3944	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
33		ffun 6673	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
34		ssid 3966	. . . . . . . 8 ⊢ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)
35		funimass2 6583	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)) → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
36	33, 34, 35	sylancl 586	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
37	24, 36	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
39		imaeq2 6016	. . . . . . 7 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (^◡𝐹 “ 𝑡)))
40	39	sseq1d 3975	. . . . . 6 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡))
41	40	rspcev 3585	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)
43	42	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡))
44	4, 43	jcad 512	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
45		elfiun 9357	. . . . . 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 584	. . . . 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 23874	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
49	5, 1, 24, 47	fmfnfmlem3 23876	. . . . . . . 8 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
50	49	eleq2d 2814	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
51	28	elrnmpt 5911	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥)))
52	51	elv 3449	. . . . . . . 8 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥))
53	5, 1, 24, 47	fmfnfmlem2 23875	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
54	52, 53	biimtrid 242	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
55	50, 54	sylbid 240	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
56	49	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
57	28	elrnmpt 5911	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
58	57	elv 3449	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥))
59	56, 58	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
60	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
61		fbssfi 23757	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑧 ∈ (fi‘𝐵)) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧)
62	5, 61	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧)
63	1	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝐿 ∈ (Fil‘𝑋))
64	1	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝐿 ∈ (Fil‘𝑋))
65	47	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
66		filtop 23775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
67	1, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋 ∈ 𝐿)
68	67, 5, 24	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
69	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
70		ssfg 23792	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
71	5, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen𝐵))
72	71	sselda 3943	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ (𝑌filGen𝐵))
73		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
74	73	imaelfm 23871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
75	69, 72, 74	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
76	65, 75	sseldd 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ 𝐿)
77	76	adantrr 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐹 “ 𝑠) ∈ 𝐿)
78	64, 77	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿))
79		filin 23774	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿 ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
80	79	3expa 1118	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
81	78, 80	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
82	81	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
83		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ⊆ 𝑋)
84		elin 3927	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) ↔ (𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥))
85	24, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → Fun 𝐹)
86		fvelima 6908	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ (𝐹 “ 𝑠)) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤)
87	86	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤))
90	85	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → Fun 𝐹)
91		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → 𝑠 ⊆ 𝑧)
92		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ 𝑠)
93		ssel2 3938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑠 ⊆ 𝑧 ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ 𝑧)
94	91, 92, 93	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ 𝑧)
95	85	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → Fun 𝐹)
96		fbelss 23753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
97	5, 96	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
98	24	fdmd 6680	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → dom 𝐹 = 𝑌)
99	98	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → dom 𝐹 = 𝑌)
100	97, 99	sseqtrrd 3981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ dom 𝐹)
101	100	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝑠 ⊆ dom 𝐹)
102	101	sselda 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ dom 𝐹)
103		fvimacnv 7007	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
104	95, 102, 103	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑦 ∈ (^◡𝐹 “ 𝑥)))
105	104	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → 𝑦 ∈ (^◡𝐹 “ 𝑥)))
106	105	impr 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
107	106	ad2ant2rl 749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
108	94, 107	elind 4159	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)))
109		inss2 4197	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ (^◡𝐹 “ 𝑥)
110		cnvimass 6042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
111	109, 110	sstri 3953	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹
112		funfvima2 7187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun 𝐹 ∧ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹) → (𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
113	111, 112	mpan2 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
116	115	expr 456	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
117		eleq1 2816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
118		eleq1 2816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ↔ 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
119	117, 118	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑤 → (((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))) ↔ (𝑤 ∈ 𝑥 → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))))
120	116, 119	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . . . . 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 3134	. . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → ((𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
124	84, 123	biimtrid 242	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
125	124	adantrl 716	. . . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
127		simprl 770	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡)
128	126, 127	sstrd 3954	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)
129		filss 23773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
130	63, 82, 83, 128, 129	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ∈ 𝐿)
131	130	exp32 420	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
132		ineq2 4173	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝑧 ∩ 𝑤) = (𝑧 ∩ (^◡𝐹 “ 𝑥)))
133	132	imaeq2d 6020	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝐹 “ (𝑧 ∩ 𝑤)) = (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
134	133	sseq1d 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 ↔ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡))
135	134	imbi1d 341	. . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → (𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
137	136	rexlimdva 3134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
138	137	rexlimdvaa 3135	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧 → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))))
139	138	imp 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
140	62, 139	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
141	60, 140	sylbid 240	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
142	141	impr 454	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (fi‘𝐵) ∧ 𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
143		imaeq2 6016	. . . . . . . . . 10 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → (𝐹 “ 𝑠) = (𝐹 “ (𝑧 ∩ 𝑤)))
144	143	sseq1d 3975	. . . . . . . . 9 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡))
145	144	imbi1d 341	. . . . . . . 8 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → (((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)) ↔ ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
146	142, 145	syl5ibrcom 247	. . . . . . 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 3192	. . . . . 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 1431	. . . . 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 240	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
150	149	rexlimdv 3132	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
151	150	impcomd 411	. 2 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡) → 𝑡 ∈ 𝐿))
152	44, 151	impbid 212	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 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 Vcvv 3444 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ↦ cmpt 5183 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 Fun wfun 6493 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ficfi 9337 fBascfbas 21284 filGencfg 21285 Filcfil 23765 FilMap cfm 23853

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1o 8411 df-2o 8412 df-en 8896 df-fin 8899 df-fi 9338 df-fbas 21293 df-fg 21294 df-fil 23766 df-fm 23858

This theorem is referenced by: fmfnfm 23878

W3C validator