fmfnfmlem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fmfnfmlem4		Structured version Visualization version GIF version

Theorem fmfnfmlem4 23468

Description: Lemma for fmfnfm 23469. (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 23363	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑡 ∈ 𝐿) → 𝑡 ⊆ 𝑋)
3	2	ex 413	. . . 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 7225	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ∈ V)
7		rnexg 7897	. . . . . . . . . . 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 7738	. . . . . . . . 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 9416	. . . . . . . . 9 ⊢ ((𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ∈ V → (𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
13	12	unssbd 4188	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ⊆ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
16		eqid 2732	. . . . . . . . 9 ⊢ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)
17		imaeq2 6055	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (^◡𝐹 “ 𝑥) = (^◡𝐹 “ 𝑡))
18	17	rspceeqv 3633	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝐿 ∧ (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑡)) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
19	16, 18	mpan2 689	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
20	19	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∃𝑥 ∈ 𝐿 (^◡𝐹 “ 𝑡) = (^◡𝐹 “ 𝑥))
21		elfvdm 6928	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
22	5, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ dom fBas)
23		cnvimass 6080	. . . . . . . . . . 11 ⊢ (^◡𝐹 “ 𝑡) ⊆ dom 𝐹
24		fmfnfm.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
25	23, 24	fssdm 6737	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ⊆ 𝑌)
26	22, 25	ssexd 5324	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 “ 𝑡) ∈ V)
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ V)
28		eqid 2732	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))
29	28	elrnmpt 5955	. . . . . . . 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 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
32	15, 31	sseldd 3983	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (^◡𝐹 “ 𝑡) ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))))
33		ffun 6720	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
34		ssid 4004	. . . . . . . 8 ⊢ (^◡𝐹 “ 𝑡) ⊆ (^◡𝐹 “ 𝑡)
35		funimass2 6631	. . . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡)
39		imaeq2 6055	. . . . . . 7 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → (𝐹 “ 𝑠) = (𝐹 “ (^◡𝐹 “ 𝑡)))
40	39	sseq1d 4013	. . . . . 6 ⊢ (𝑠 = (^◡𝐹 “ 𝑡) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (^◡𝐹 “ 𝑡)) ⊆ 𝑡))
41	40	rspcev 3612	. . . . 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 413	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡))
44	4, 43	jcad 513	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐿 → (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡)))
45		elfiun 9427	. . . . . 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 23465	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
49	5, 1, 24, 47	fmfnfmlem3 23467	. . . . . . . 8 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))
50	49	eleq2d 2819	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
51	28	elrnmpt 5955	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥)))
52	51	elv 3480	. . . . . . . 8 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥))
53	5, 1, 24, 47	fmfnfmlem2 23466	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (^◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
54	52, 53	biimtrid 241	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
55	50, 54	sylbid 239	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
56	49	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ 𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))
57	28	elrnmpt 5955	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
58	57	elv 3480	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥))
59	56, 58	bitrdi 286	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
60	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) ↔ ∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥)))
61		fbssfi 23348	. . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝐿 ∈ (Fil‘𝑋))
64	1	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝐿 ∈ (Fil‘𝑋))
65	47	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
66		filtop 23366	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋))
70		ssfg 23383	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
71	5, 70	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 ⊆ (𝑌filGen𝐵))
72	71	sselda 3982	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ∈ (𝑌filGen𝐵))
73		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
74	73	imaelfm 23462	. . . . . . . . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝐹 “ 𝑠) ∈ 𝐿)
77	76	adantrr 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐹 “ 𝑠) ∈ 𝐿)
78	64, 77	jca 512	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (𝐿 ∈ (Fil‘𝑋) ∧ (𝐹 “ 𝑠) ∈ 𝐿))
79		filin 23365	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿)
83		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ⊆ 𝑋)
84		elin 3964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) ↔ (𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥))
85	24, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → Fun 𝐹)
86		fvelima 6957	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑤 ∈ (𝐹 “ 𝑠)) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤)
87	86	ex 413	. . . . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ (𝐹 “ 𝑠) → ∃𝑦 ∈ 𝑠 (𝐹‘𝑦) = 𝑤))
90	85	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → Fun 𝐹)
91		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → 𝑠 ⊆ 𝑧)
92		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ 𝑠)
93		ssel2 3977	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → Fun 𝐹)
96		fbelss 23344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
97	5, 96	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ 𝑌)
98	24	fdmd 6728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → dom 𝐹 = 𝑌)
99	98	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → dom 𝐹 = 𝑌)
100	97, 99	sseqtrrd 4023	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → 𝑠 ⊆ dom 𝐹)
101	100	adantrr 715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → 𝑠 ⊆ dom 𝐹)
102	101	sselda 3982	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → 𝑦 ∈ dom 𝐹)
103		fvimacnv 7054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → 𝑦 ∈ (^◡𝐹 “ 𝑥)))
106	105	impr 455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
107	106	ad2ant2rl 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (^◡𝐹 “ 𝑥))
108	94, 107	elind 4194	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ (𝑡 ⊆ 𝑋 ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥))) → 𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)))
109		inss2 4229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ (^◡𝐹 “ 𝑥)
110		cnvimass 6080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (^◡𝐹 “ 𝑥) ⊆ dom 𝐹
111	109, 110	sstri 3991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹
112		funfvima2 7235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun 𝐹 ∧ (𝑧 ∩ (^◡𝐹 “ 𝑥)) ⊆ dom 𝐹) → (𝑦 ∈ (𝑧 ∩ (^◡𝐹 “ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
113	111, 112	mpan2 689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ (𝑦 ∈ 𝑠 ∧ (𝐹‘𝑦) ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
116	115	expr 457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑠) → ((𝐹‘𝑦) ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
117		eleq1 2821	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹‘𝑦) = 𝑤 → ((𝐹‘𝑦) ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
118		eleq1 2821	. . . . . . . . . . . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → ((𝑤 ∈ (𝐹 “ 𝑠) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
124	84, 123	biimtrid 241	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ 𝑡 ⊆ 𝑋) → (𝑤 ∈ ((𝐹 “ 𝑠) ∩ 𝑥) → 𝑤 ∈ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥)))))
125	124	adantrl 714	. . . . . . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
127		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡)
128	126, 127	sstrd 3992	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)
129		filss 23364	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (((𝐹 “ 𝑠) ∩ 𝑥) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ ((𝐹 “ 𝑠) ∩ 𝑥) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
130	63, 82, 83, 128, 129	syl13anc 1372	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ∈ 𝐿)
131	130	exp32 421	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
132		ineq2 4206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝑧 ∩ 𝑤) = (𝑧 ∩ (^◡𝐹 “ 𝑥)))
133	132	imaeq2d 6059	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (^◡𝐹 “ 𝑥) → (𝐹 “ (𝑧 ∩ 𝑤)) = (𝐹 “ (𝑧 ∩ (^◡𝐹 “ 𝑥))))
134	133	sseq1d 4013	. . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) ∧ 𝑥 ∈ 𝐿) → (𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
137	136	rexlimdva 3155	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑠 ⊆ 𝑧)) → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
138	137	rexlimdvaa 3156	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑠 ∈ 𝐵 𝑠 ⊆ 𝑧 → (∃𝑥 ∈ 𝐿 𝑤 = (^◡𝐹 “ 𝑥) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))))
139	138	imp 407	. . . . . . . . . . 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 239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (fi‘𝐵)) → (𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
142	141	impr 455	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (fi‘𝐵) ∧ 𝑤 ∈ (fi‘ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))) → ((𝐹 “ (𝑧 ∩ 𝑤)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
143		imaeq2 6055	. . . . . . . . . 10 ⊢ (𝑠 = (𝑧 ∩ 𝑤) → (𝐹 “ 𝑠) = (𝐹 “ (𝑧 ∩ 𝑤)))
144	143	sseq1d 4013	. . . . . . . . 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 246	. . . . . . 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 3211	. . . . . 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 1428	. . . . 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 239	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥)))) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
150	149	rexlimdv 3153	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
151	150	impcomd 412	. 2 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (^◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡) → 𝑡 ∈ 𝐿))
152	44, 151	impbid 211	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 205 ∧ wa 396 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ↦ cmpt 5231 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ficfi 9407 fBascfbas 20938 filGencfg 20939 Filcfil 23356 FilMap cfm 23444

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1o 8468 df-er 8705 df-en 8942 df-fin 8945 df-fi 9408 df-fbas 20947 df-fg 20948 df-fil 23357 df-fm 23449

This theorem is referenced by: fmfnfm 23469

W3C validator