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Theorem fmfnfmlem3 22088
Description: Lemma for fmfnfm 22090. (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 21986 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦𝐿𝑧𝐿) → (𝑦𝑧) ∈ 𝐿)
323expb 1150 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
41, 3sylan 576 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
5 fmfnfm.f . . . . . . . . 9 (𝜑𝐹:𝑌𝑋)
6 ffun 6259 . . . . . . . . 9 (𝐹:𝑌𝑋 → Fun 𝐹)
7 funcnvcnv 6167 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
8 imain 6185 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑦𝑧)) = ((𝐹𝑦) ∩ (𝐹𝑧)))
98eqcomd 2805 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
105, 6, 7, 94syl 19 . . . . . . . 8 (𝜑 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
1110adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
12 imaeq2 5679 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝐹𝑥) = (𝐹 “ (𝑦𝑧)))
1312rspceeqv 3515 . . . . . . 7 (((𝑦𝑧) ∈ 𝐿 ∧ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
144, 11, 13syl2anc 580 . . . . . 6 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
15 ineq12 4007 . . . . . . . 8 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (𝑠𝑡) = ((𝐹𝑦) ∩ (𝐹𝑧)))
1615eqeq1d 2801 . . . . . . 7 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ((𝑠𝑡) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1716rexbidv 3233 . . . . . 6 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥) ↔ ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1814, 17syl5ibrcom 239 . . . . 5 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
1918rexlimdvva 3219 . . . 4 (𝜑 → (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
20 imaeq2 5679 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2120eqeq2d 2809 . . . . . . 7 (𝑥 = 𝑦 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑦)))
2221cbvrexv 3355 . . . . . 6 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑦𝐿 𝑠 = (𝐹𝑦))
23 imaeq2 5679 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2423eqeq2d 2809 . . . . . . 7 (𝑥 = 𝑧 → (𝑡 = (𝐹𝑥) ↔ 𝑡 = (𝐹𝑧)))
2524cbvrexv 3355 . . . . . 6 (∃𝑥𝐿 𝑡 = (𝐹𝑥) ↔ ∃𝑧𝐿 𝑡 = (𝐹𝑧))
2622, 25anbi12i 621 . . . . 5 ((∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
27 vex 3388 . . . . . . 7 𝑠 ∈ V
28 eqid 2799 . . . . . . . 8 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
2928elrnmpt 5576 . . . . . . 7 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
3027, 29ax-mp 5 . . . . . 6 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
31 vex 3388 . . . . . . 7 𝑡 ∈ V
3228elrnmpt 5576 . . . . . . 7 (𝑡 ∈ V → (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
3331, 32ax-mp 5 . . . . . 6 (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥))
3430, 33anbi12i 621 . . . . 5 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
35 reeanv 3288 . . . . 5 (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
3626, 34, 353bitr4i 295 . . . 4 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)))
3727inex1 4994 . . . . 5 (𝑠𝑡) ∈ V
3828elrnmpt 5576 . . . . 5 ((𝑠𝑡) ∈ V → ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
3937, 38ax-mp 5 . . . 4 ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥))
4019, 36, 393imtr4g 288 . . 3 (𝜑 → ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → (𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))))
4140ralrimivv 3151 . 2 (𝜑 → ∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
42 mptexg 6713 . . 3 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
43 rnexg 7332 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
44 inficl 8573 . . 3 (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
451, 42, 43, 444syl 19 . 2 (𝜑 → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
4641, 45mpbid 224 1 (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  cin 3768  wss 3769  cmpt 4922  ccnv 5311  ran crn 5313  cima 5315  Fun wfun 6095  wf 6097  cfv 6101  (class class class)co 6878  ficfi 8558  fBascfbas 20056  Filcfil 21977   FilMap cfm 22065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-fin 8199  df-fi 8559  df-fbas 20065  df-fil 21978
This theorem is referenced by:  fmfnfmlem4  22089
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