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Theorem fmfnfmlem3 21961
Description: Lemma for fmfnfm 21963. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypotheses
Ref Expression
fmfnfm.b (𝜑𝐵 ∈ (fBas‘𝑌))
fmfnfm.l (𝜑𝐿 ∈ (Fil‘𝑋))
fmfnfm.f (𝜑𝐹:𝑌𝑋)
fmfnfm.fm (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)
Assertion
Ref Expression
fmfnfmlem3 (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))
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.
StepHypRef Expression
1 fmfnfm.l . . . . . . . 8 (𝜑𝐿 ∈ (Fil‘𝑋))
2 filin 21859 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦𝐿𝑧𝐿) → (𝑦𝑧) ∈ 𝐿)
323expb 1114 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
41, 3sylan 489 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
5 fmfnfm.f . . . . . . . . 9 (𝜑𝐹:𝑌𝑋)
6 ffun 6209 . . . . . . . . 9 (𝐹:𝑌𝑋 → Fun 𝐹)
7 funcnvcnv 6117 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
8 imain 6135 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑦𝑧)) = ((𝐹𝑦) ∩ (𝐹𝑧)))
98eqcomd 2766 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
105, 6, 7, 94syl 19 . . . . . . . 8 (𝜑 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
1110adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
12 imaeq2 5620 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐹𝑥) = (𝐹 “ (𝑦𝑧)))
1312eqeq2d 2770 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))))
1413rspcev 3449 . . . . . . 7 (((𝑦𝑧) ∈ 𝐿 ∧ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
154, 11, 14syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
16 ineq12 3952 . . . . . . . 8 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (𝑠𝑡) = ((𝐹𝑦) ∩ (𝐹𝑧)))
1716eqeq1d 2762 . . . . . . 7 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ((𝑠𝑡) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1817rexbidv 3190 . . . . . 6 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥) ↔ ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1915, 18syl5ibrcom 237 . . . . 5 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
2019rexlimdvva 3176 . . . 4 (𝜑 → (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
21 imaeq2 5620 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2221eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑦 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑦)))
2322cbvrexv 3311 . . . . . 6 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑦𝐿 𝑠 = (𝐹𝑦))
24 imaeq2 5620 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2524eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑧 → (𝑡 = (𝐹𝑥) ↔ 𝑡 = (𝐹𝑧)))
2625cbvrexv 3311 . . . . . 6 (∃𝑥𝐿 𝑡 = (𝐹𝑥) ↔ ∃𝑧𝐿 𝑡 = (𝐹𝑧))
2723, 26anbi12i 735 . . . . 5 ((∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
28 vex 3343 . . . . . . 7 𝑠 ∈ V
29 eqid 2760 . . . . . . . 8 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
3029elrnmpt 5527 . . . . . . 7 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
3128, 30ax-mp 5 . . . . . 6 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
32 vex 3343 . . . . . . 7 𝑡 ∈ V
3329elrnmpt 5527 . . . . . . 7 (𝑡 ∈ V → (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
3432, 33ax-mp 5 . . . . . 6 (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥))
3531, 34anbi12i 735 . . . . 5 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
36 reeanv 3245 . . . . 5 (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
3727, 35, 363bitr4i 292 . . . 4 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)))
3828inex1 4951 . . . . 5 (𝑠𝑡) ∈ V
3929elrnmpt 5527 . . . . 5 ((𝑠𝑡) ∈ V → ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
4038, 39ax-mp 5 . . . 4 ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥))
4120, 37, 403imtr4g 285 . . 3 (𝜑 → ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → (𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))))
4241ralrimivv 3108 . 2 (𝜑 → ∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
43 mptexg 6648 . . 3 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
44 rnexg 7263 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
45 inficl 8496 . . 3 (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
461, 43, 44, 454syl 19 . 2 (𝜑 → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
4742, 46mpbid 222 1 (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  cmpt 4881  ccnv 5265  ran crn 5267  cima 5269  Fun wfun 6043  wf 6045  cfv 6049  (class class class)co 6813  ficfi 8481  fBascfbas 19936  Filcfil 21850   FilMap cfm 21938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-fbas 19945  df-fil 21851
This theorem is referenced by:  fmfnfmlem4  21962
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