Theorem fmfnfmlem3 23943

Description: Lemma for fmfnfm 23945. (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 23841	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿) → (𝑦 ∩ 𝑧) ∈ 𝐿)
3	2	3expb 1127	. . . . . . . 8 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
4	1, 3	sylan 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
5		fmfnfm.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffun 6662	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
7		funcnvcnv 6556	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
8		imain 6574	. . . . . . . . . 10 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑦 ∩ 𝑧)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
9	8	eqcomd 2747	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
12		imaeq2 6015	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (◡𝐹 “ 𝑥) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
13	12	rspceeqv 3585	. . . . . . 7 ⊢ (((𝑦 ∩ 𝑧) ∈ 𝐿 ∧ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧))) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
14	4, 11, 13	syl2anc 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
15		ineq12 4147	. . . . . . . 8 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → (𝑠 ∩ 𝑡) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
16	15	eqeq1d 2743	. . . . . . 7 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ((𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
17	16	rexbidv 3165	. . . . . 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 3198	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐿 ∃𝑧 ∈ 𝐿 (𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
20		imaeq2 6015	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
21	20	eqeq2d 2752	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑦)))
22	21	cbvrexvw 3220	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦))
23		imaeq2 6015	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑧))
24	23	eqeq2d 2752	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑡 = (◡𝐹 “ 𝑥) ↔ 𝑡 = (◡𝐹 “ 𝑧)))
25	24	cbvrexvw 3220	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥) ↔ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧))
26	22, 25	anbi12i 635	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ∧ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)) ↔ (∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦) ∧ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧)))
27		eqid 2741	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
28	27	elrnmpt 5907	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
29	28	elv 3438	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
30	27	elrnmpt 5907	. . . . . . 7 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)))
31	30	elv 3438	. . . . . 6 ⊢ (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥))
32	29, 31	anbi12i 635	. . . . 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 3213	. . . . 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 3437	. . . . . 6 ⊢ 𝑠 ∈ V
36	35	inex1 5248	. . . . 5 ⊢ (𝑠 ∩ 𝑡) ∈ V
37	27	elrnmpt 5907	. . . . 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 3182	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
41		mptexg 7169	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
42		rnexg 7846	. . 3 ⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
43		inficl 9332	. . 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 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ∩ cin 3884   ⊆ wss 3885   ↦ cmpt 5156  ^◡ccnv 5620  ran crn 5622   “ cima 5624  Fun wfun 6483  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  ficfi 9317  fBascfbas 21339  Filcfil 23832   FilMap cfm 23920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-2o 8400  df-en 8888  df-fin 8891  df-fi 9318  df-fbas 21348  df-fil 23833

This theorem is referenced by:  fmfnfmlem4  23944