Theorem fmfnfmlem3 22566

Description: Lemma for fmfnfm 22568. (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
fmfnfmlem3	⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))

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

Proof of Theorem fmfnfmlem3

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

Step	Hyp	Ref	Expression
1		fmfnfm.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋))
2		filin 22464	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿) → (𝑦 ∩ 𝑧) ∈ 𝐿)
3	2	3expb 1116	. . . . . . . 8 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
4	1, 3	sylan 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
5		fmfnfm.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffun 6519	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
7		funcnvcnv 6423	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
8		imain 6441	. . . . . . . . . 10 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑦 ∩ 𝑧)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
9	8	eqcomd 2829	. . . . . . . . 9 ⊢ (Fun ◡◡𝐹 → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
10	5, 6, 7, 9	4syl 19	. . . . . . . 8 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
11	10	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
12		imaeq2 5927	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (◡𝐹 “ 𝑥) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
13	12	rspceeqv 3640	. . . . . . 7 ⊢ (((𝑦 ∩ 𝑧) ∈ 𝐿 ∧ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧))) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
14	4, 11, 13	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
15		ineq12 4186	. . . . . . . 8 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → (𝑠 ∩ 𝑡) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
16	15	eqeq1d 2825	. . . . . . 7 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ((𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
17	16	rexbidv 3299	. . . . . 6 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → (∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
18	14, 17	syl5ibrcom 249	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
19	18	rexlimdvva 3296	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐿 ∃𝑧 ∈ 𝐿 (𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
20		imaeq2 5927	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
21	20	eqeq2d 2834	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑦)))
22	21	cbvrexvw 3452	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦))
23		imaeq2 5927	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑧))
24	23	eqeq2d 2834	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑡 = (◡𝐹 “ 𝑥) ↔ 𝑡 = (◡𝐹 “ 𝑧)))
25	24	cbvrexvw 3452	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥) ↔ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧))
26	22, 25	anbi12i 628	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ∧ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)) ↔ (∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦) ∧ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧)))
27		eqid 2823	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
28	27	elrnmpt 5830	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
29	28	elv 3501	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
30	27	elrnmpt 5830	. . . . . . 7 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)))
31	30	elv 3501	. . . . . 6 ⊢ (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥))
32	29, 31	anbi12i 628	. . . . 5 ⊢ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ∧ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)))
33		reeanv 3369	. . . . 5 ⊢ (∃𝑦 ∈ 𝐿 ∃𝑧 ∈ 𝐿 (𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) ↔ (∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦) ∧ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧)))
34	26, 32, 33	3bitr4i 305	. . . 4 ⊢ ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) ↔ ∃𝑦 ∈ 𝐿 ∃𝑧 ∈ 𝐿 (𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)))
35		vex 3499	. . . . . 6 ⊢ 𝑠 ∈ V
36	35	inex1 5223	. . . . 5 ⊢ (𝑠 ∩ 𝑡) ∈ V
37	27	elrnmpt 5830	. . . . 5 ⊢ ((𝑠 ∩ 𝑡) ∈ V → ((𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
38	36, 37	ax-mp 5	. . . 4 ⊢ ((𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥))
39	19, 34, 38	3imtr4g 298	. . 3 ⊢ (𝜑 → ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) → (𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
40	39	ralrimivv 3192	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
41		mptexg 6986	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
42		rnexg 7616	. . 3 ⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
43		inficl 8891	. . 3 ⊢ (ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → (∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
44	1, 41, 42, 43	4syl 19	. 2 ⊢ (𝜑 → (∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
45	40, 44	mpbid 234	1 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938   ↦ cmpt 5148  ^◡ccnv 5556  ran crn 5558   “ cima 5560  Fun wfun 6351  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ficfi 8876  fBascfbas 20535  Filcfil 22455   FilMap cfm 22543

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-fin 8515  df-fi 8877  df-fbas 20544  df-fil 22456

This theorem is referenced by:  fmfnfmlem4  22567