Theorem fmfnfmlem3 23152

Description: Lemma for fmfnfm 23154. (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 23050	. . . . . . . . 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 581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
5		fmfnfm.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffun 6633	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
7		funcnvcnv 6530	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
8		imain 6548	. . . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
12		imaeq2 5975	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (◡𝐹 “ 𝑥) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
13	12	rspceeqv 3580	. . . . . . 7 ⊢ (((𝑦 ∩ 𝑧) ∈ 𝐿 ∧ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧))) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
14	4, 11, 13	syl2anc 585	. . . . . 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 2738	. . . . . . 7 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ((𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
17	16	rexbidv 3172	. . . . . 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 3202	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐿 ∃𝑧 ∈ 𝐿 (𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ∃𝑥 ∈ 𝐿 (𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥)))
20		imaeq2 5975	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
21	20	eqeq2d 2747	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑦)))
22	21	cbvrexvw 3223	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦))
23		imaeq2 5975	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑧))
24	23	eqeq2d 2747	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑡 = (◡𝐹 “ 𝑥) ↔ 𝑡 = (◡𝐹 “ 𝑧)))
25	24	cbvrexvw 3223	. . . . . 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 5877	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
29	28	elv 3443	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
30	27	elrnmpt 5877	. . . . . . 7 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)))
31	30	elv 3443	. . . . . 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 3214	. . . . 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 3441	. . . . . 6 ⊢ 𝑠 ∈ V
36	35	inex1 5250	. . . . 5 ⊢ (𝑠 ∩ 𝑡) ∈ V
37	27	elrnmpt 5877	. . . . 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 3192	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
41		mptexg 7129	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
42		rnexg 7783	. . 3 ⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
43		inficl 9228	. . 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 231	1 ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071  Vcvv 3437   ∩ cin 3891   ⊆ wss 3892   ↦ cmpt 5164  ^◡ccnv 5599  ran crn 5601   “ cima 5603  Fun wfun 6452  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307  ficfi 9213  fBascfbas 20630  Filcfil 23041   FilMap cfm 23129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-om 7745  df-1o 8328  df-er 8529  df-en 8765  df-fin 8768  df-fi 9214  df-fbas 20639  df-fil 23042

This theorem is referenced by:  fmfnfmlem4  23153