Theorem fmfnfmlem3 23900

Description: Lemma for fmfnfm 23902. (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 23798	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿) → (𝑦 ∩ 𝑧) ∈ 𝐿)
3	2	3expb 1120	. . . . . . . 8 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
4	1, 3	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
5		fmfnfm.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffun 6665	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
7		funcnvcnv 6559	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
8		imain 6577	. . . . . . . . . 10 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑦 ∩ 𝑧)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
9	8	eqcomd 2742	. . . . . . . . 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 480	. . . . . . 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 3599	. . . . . . 7 ⊢ (((𝑦 ∩ 𝑧) ∈ 𝐿 ∧ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧))) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
14	4, 11, 13	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
15		ineq12 4167	. . . . . . . 8 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → (𝑠 ∩ 𝑡) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
16	15	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ((𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
17	16	rexbidv 3160	. . . . . 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 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
19	18	rexlimdvva 3193	. . . 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 2747	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑦)))
22	21	cbvrexvw 3215	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦))
23		imaeq2 6015	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑧))
24	23	eqeq2d 2747	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑡 = (◡𝐹 “ 𝑥) ↔ 𝑡 = (◡𝐹 “ 𝑧)))
25	24	cbvrexvw 3215	. . . . . 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 2736	. . . . . . . 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 3445	. . . . . 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 3445	. . . . . 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 3208	. . . . 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 303	. . . 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 3444	. . . . . 6 ⊢ 𝑠 ∈ V
36	35	inex1 5262	. . . . 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 296	. . 3 ⊢ (𝜑 → ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) → (𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
40	39	ralrimivv 3177	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
41		mptexg 7167	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
42		rnexg 7844	. . 3 ⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
43		inficl 9328	. . 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 232	1 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   ↦ cmpt 5179  ^◡ccnv 5623  ran crn 5625   “ cima 5627  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ficfi 9313  fBascfbas 21297  Filcfil 23789   FilMap cfm 23877

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-2o 8398  df-en 8884  df-fin 8887  df-fi 9314  df-fbas 21306  df-fil 23790

This theorem is referenced by:  fmfnfmlem4  23901