Theorem fmfnfmlem2 23680

Description: Lemma for fmfnfm 23683. (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
fmfnfmlem2	⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))

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 fmfnfmlem2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmfnfm.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋))
2	1	ad2antrr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝐿 ∈ (Fil‘𝑋))
3		simplr 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑥 ∈ 𝐿)
4		fmfnfm.fm	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
5		fmfnfm.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffn 6718	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
7		dffn4 6812	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑌 ↔ 𝐹:𝑌–onto→ran 𝐹)
8	6, 7	sylib 217	. . . . . . . . . 10 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌–onto→ran 𝐹)
9		foima 6811	. . . . . . . . . 10 ⊢ (𝐹:𝑌–onto→ran 𝐹 → (𝐹 “ 𝑌) = ran 𝐹)
10	5, 8, 9	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) = ran 𝐹)
11		filtop 23580	. . . . . . . . . . 11 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿)
12	1, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐿)
13		fmfnfm.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌))
14		fgcl 23603	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
15		filtop 23580	. . . . . . . . . . 11 ⊢ ((𝑌filGen𝐵) ∈ (Fil‘𝑌) → 𝑌 ∈ (𝑌filGen𝐵))
16	13, 14, 15	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑌filGen𝐵))
17		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑌filGen𝐵) = (𝑌filGen𝐵)
18	17	imaelfm 23676	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑌 ∈ (𝑌filGen𝐵)) → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
19	12, 13, 5, 16, 18	syl31anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑌) ∈ ((𝑋 FilMap 𝐹)‘𝐵))
20	10, 19	eqeltrrd 2833	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ∈ ((𝑋 FilMap 𝐹)‘𝐵))
21	4, 20	sseldd 3984	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ∈ 𝐿)
22	21	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ran 𝐹 ∈ 𝐿)
23		filin 23579	. . . . . 6 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
24	2, 3, 22, 23	syl3anc 1370	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑥 ∩ ran 𝐹) ∈ 𝐿)
25		simprr 770	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ⊆ 𝑋)
26		elin 3965	. . . . . . 7 ⊢ (𝑦 ∈ (𝑥 ∩ ran 𝐹) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ran 𝐹))
27		fvelrnb 6953	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑌 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦))
28	5, 6, 27	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦))
29	28	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦))
30	5	ffund 6722	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Fun 𝐹)
31	30	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → Fun 𝐹)
32		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → 𝑧 ∈ 𝑌)
33	5	fdmd 6729	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝑌)
34	33	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → dom 𝐹 = 𝑌)
35	32, 34	eleqtrrd 2835	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → 𝑧 ∈ dom 𝐹)
36		fvimacnv 7055	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝑧 ∈ dom 𝐹) → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑧 ∈ (◡𝐹 “ 𝑥)))
37	31, 35, 36	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑧 ∈ (◡𝐹 “ 𝑥)))
38		cnvimass 6081	. . . . . . . . . . . . . . . 16 ⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹
39		funfvima2 7236	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝑥) ⊆ dom 𝐹) → (𝑧 ∈ (◡𝐹 “ 𝑥) → (𝐹‘𝑧) ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
40	31, 38, 39	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → (𝑧 ∈ (◡𝐹 “ 𝑥) → (𝐹‘𝑧) ∈ (𝐹 “ (◡𝐹 “ 𝑥))))
41		ssel 3976	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 → ((𝐹‘𝑧) ∈ (𝐹 “ (◡𝐹 “ 𝑥)) → (𝐹‘𝑧) ∈ 𝑡))
42	41	ad2antrl 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) ∈ (𝐹 “ (◡𝐹 “ 𝑥)) → (𝐹‘𝑧) ∈ 𝑡))
43	40, 42	syld 47	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → (𝑧 ∈ (◡𝐹 “ 𝑥) → (𝐹‘𝑧) ∈ 𝑡))
44	37, 43	sylbid 239	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) ∈ 𝑥 → (𝐹‘𝑧) ∈ 𝑡))
45		eleq1 2820	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑧) = 𝑦 → ((𝐹‘𝑧) ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
46		eleq1 2820	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑧) = 𝑦 → ((𝐹‘𝑧) ∈ 𝑡 ↔ 𝑦 ∈ 𝑡))
47	45, 46	imbi12d 343	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = 𝑦 → (((𝐹‘𝑧) ∈ 𝑥 → (𝐹‘𝑧) ∈ 𝑡) ↔ (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑡)))
48	44, 47	syl5ibcom 244	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = 𝑦 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑡)))
49	48	expr 456	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡) → (𝑧 ∈ 𝑌 → ((𝐹‘𝑧) = 𝑦 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑡))))
50	49	rexlimdv 3152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = 𝑦 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑡)))
51	29, 50	sylbid 239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡) → (𝑦 ∈ ran 𝐹 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑡)))
52	51	impcomd 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡) → ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ 𝑡))
53	52	adantrr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ 𝑡))
54	26, 53	biimtrid 241	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑦 ∈ (𝑥 ∩ ran 𝐹) → 𝑦 ∈ 𝑡))
55	54	ssrdv 3989	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → (𝑥 ∩ ran 𝐹) ⊆ 𝑡)
56		filss 23578	. . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ ((𝑥 ∩ ran 𝐹) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ (𝑥 ∩ ran 𝐹) ⊆ 𝑡)) → 𝑡 ∈ 𝐿)
57	2, 24, 25, 55, 56	syl13anc 1371	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐿) ∧ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋)) → 𝑡 ∈ 𝐿)
58	57	exp32 420	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐿) → ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))
59		imaeq2 6056	. . . . 5 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (𝐹 “ 𝑠) = (𝐹 “ (◡𝐹 “ 𝑥)))
60	59	sseq1d 4014	. . . 4 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 ↔ (𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡))
61	60	imbi1d 340	. . 3 ⊢ (𝑠 = (◡𝐹 “ 𝑥) → (((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)) ↔ ((𝐹 “ (◡𝐹 “ 𝑥)) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
62	58, 61	syl5ibrcom 246	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐿) → (𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))
63	62	rexlimdva 3154	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∃wrex 3069   ∩ cin 3948   ⊆ wss 3949  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   “ cima 5680  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  –onto→wfo 6542  ‘cfv 6544  (class class class)co 7412  fBascfbas 21133  filGencfg 21134  Filcfil 23570   FilMap cfm 23658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21142  df-fg 21143  df-fil 23571  df-fm 23663

This theorem is referenced by:  fmfnfmlem4  23682