Theorem fmfnfmlem3 23876

Description: Lemma for fmfnfm 23878. (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 23774	. . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿) → (𝑦 ∩ 𝑧) ∈ 𝐿)
3	2	3expb 1117	. . . . . . . 8 ⊢ ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
4	1, 3	sylan 578	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → (𝑦 ∩ 𝑧) ∈ 𝐿)
5		fmfnfm.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋)
6		ffun 6719	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → Fun 𝐹)
7		funcnvcnv 6614	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
8		imain 6632	. . . . . . . . . 10 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝑦 ∩ 𝑧)) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
9	8	eqcomd 2731	. . . . . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
12		imaeq2 6054	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (◡𝐹 “ 𝑥) = (◡𝐹 “ (𝑦 ∩ 𝑧)))
13	12	rspceeqv 3624	. . . . . . 7 ⊢ (((𝑦 ∩ 𝑧) ∈ 𝐿 ∧ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ (𝑦 ∩ 𝑧))) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
14	4, 11, 13	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐿 ∧ 𝑧 ∈ 𝐿)) → ∃𝑥 ∈ 𝐿 ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥))
15		ineq12 4201	. . . . . . . 8 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → (𝑠 ∩ 𝑡) = ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)))
16	15	eqeq1d 2727	. . . . . . 7 ⊢ ((𝑠 = (◡𝐹 “ 𝑦) ∧ 𝑡 = (◡𝐹 “ 𝑧)) → ((𝑠 ∩ 𝑡) = (◡𝐹 “ 𝑥) ↔ ((◡𝐹 “ 𝑦) ∩ (◡𝐹 “ 𝑧)) = (◡𝐹 “ 𝑥)))
17	16	rexbidv 3169	. . . . . 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 246	. . . . 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 6054	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑦))
21	20	eqeq2d 2736	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑠 = (◡𝐹 “ 𝑥) ↔ 𝑠 = (◡𝐹 “ 𝑦)))
22	21	cbvrexvw 3226	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ↔ ∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦))
23		imaeq2 6054	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝑧))
24	23	eqeq2d 2736	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑡 = (◡𝐹 “ 𝑥) ↔ 𝑡 = (◡𝐹 “ 𝑧)))
25	24	cbvrexvw 3226	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥) ↔ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧))
26	22, 25	anbi12i 626	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) ∧ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)) ↔ (∃𝑦 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑦) ∧ ∃𝑧 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑧)))
27		eqid 2725	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) = (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))
28	27	elrnmpt 5952	. . . . . . 7 ⊢ (𝑠 ∈ V → (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥)))
29	28	elv 3469	. . . . . 6 ⊢ (𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥))
30	27	elrnmpt 5952	. . . . . . 7 ⊢ (𝑡 ∈ V → (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥)))
31	30	elv 3469	. . . . . 6 ⊢ (𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ↔ ∃𝑥 ∈ 𝐿 𝑡 = (◡𝐹 “ 𝑥))
32	29, 31	anbi12i 626	. . . . 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 3217	. . . . 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 302	. . . 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 3467	. . . . . 6 ⊢ 𝑠 ∈ V
36	35	inex1 5312	. . . . 5 ⊢ (𝑠 ∩ 𝑡) ∈ V
37	27	elrnmpt 5952	. . . . 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 295	. . 3 ⊢ (𝜑 → ((𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∧ 𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) → (𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))
40	39	ralrimivv 3189	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))∀𝑡 ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))(𝑠 ∩ 𝑡) ∈ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)))
41		mptexg 7228	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
42		rnexg 7906	. . 3 ⊢ ((𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ V)
43		inficl 9446	. . 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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ∩ cin 3939   ⊆ wss 3940   ↦ cmpt 5226  ^◡ccnv 5671  ran crn 5673   “ cima 5675  Fun wfun 6536  ⟶wf 6538  ‘cfv 6542  (class class class)co 7415  ficfi 9431  fBascfbas 21269  Filcfil 23765   FilMap cfm 23853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7868  df-1o 8483  df-er 8721  df-en 8961  df-fin 8964  df-fi 9432  df-fbas 21278  df-fil 23766

This theorem is referenced by:  fmfnfmlem4  23877