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Theorem fmfnfmlem3 21932
Description: Lemma for fmfnfm 21934. (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 21830 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦𝐿𝑧𝐿) → (𝑦𝑧) ∈ 𝐿)
323expb 1113 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
41, 3sylan 489 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
5 fmfnfm.f . . . . . . . . 9 (𝜑𝐹:𝑌𝑋)
6 ffun 6197 . . . . . . . . 9 (𝐹:𝑌𝑋 → Fun 𝐹)
7 funcnvcnv 6105 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
8 imain 6123 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑦𝑧)) = ((𝐹𝑦) ∩ (𝐹𝑧)))
98eqcomd 2754 . . . . . . . . 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 5608 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐹𝑥) = (𝐹 “ (𝑦𝑧)))
1312eqeq2d 2758 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))))
1413rspcev 3437 . . . . . . 7 (((𝑦𝑧) ∈ 𝐿 ∧ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
154, 11, 14syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
16 ineq12 3940 . . . . . . . 8 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (𝑠𝑡) = ((𝐹𝑦) ∩ (𝐹𝑧)))
1716eqeq1d 2750 . . . . . . 7 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ((𝑠𝑡) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1817rexbidv 3178 . . . . . 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 3164 . . . 4 (𝜑 → (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
21 imaeq2 5608 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2221eqeq2d 2758 . . . . . . 7 (𝑥 = 𝑦 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑦)))
2322cbvrexv 3299 . . . . . 6 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑦𝐿 𝑠 = (𝐹𝑦))
24 imaeq2 5608 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2524eqeq2d 2758 . . . . . . 7 (𝑥 = 𝑧 → (𝑡 = (𝐹𝑥) ↔ 𝑡 = (𝐹𝑧)))
2625cbvrexv 3299 . . . . . 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 3331 . . . . . . 7 𝑠 ∈ V
29 eqid 2748 . . . . . . . 8 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
3029elrnmpt 5515 . . . . . . 7 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
3128, 30ax-mp 5 . . . . . 6 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
32 vex 3331 . . . . . . 7 𝑡 ∈ V
3329elrnmpt 5515 . . . . . . 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 3233 . . . . 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 4939 . . . . 5 (𝑠𝑡) ∈ V
3929elrnmpt 5515 . . . . 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 3096 . 2 (𝜑 → ∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
43 mptexg 6636 . . 3 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
44 rnexg 7251 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
45 inficl 8484 . . 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 1620  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  cin 3702  wss 3703  cmpt 4869  ccnv 5253  ran crn 5255  cima 5257  Fun wfun 6031  wf 6033  cfv 6037  (class class class)co 6801  ficfi 8469  fBascfbas 19907  Filcfil 21821   FilMap cfm 21909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-en 8110  df-fin 8113  df-fi 8470  df-fbas 19916  df-fil 21822
This theorem is referenced by:  fmfnfmlem4  21933
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