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Theorem fmfnfmlem3 22671
 Description: Lemma for fmfnfm 22673. (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 22569 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦𝐿𝑧𝐿) → (𝑦𝑧) ∈ 𝐿)
323expb 1118 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
41, 3sylan 583 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
5 fmfnfm.f . . . . . . . . 9 (𝜑𝐹:𝑌𝑋)
6 ffun 6507 . . . . . . . . 9 (𝐹:𝑌𝑋 → Fun 𝐹)
7 funcnvcnv 6408 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
8 imain 6426 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑦𝑧)) = ((𝐹𝑦) ∩ (𝐹𝑧)))
98eqcomd 2765 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
105, 6, 7, 94syl 19 . . . . . . . 8 (𝜑 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
1110adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
12 imaeq2 5903 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝐹𝑥) = (𝐹 “ (𝑦𝑧)))
1312rspceeqv 3559 . . . . . . 7 (((𝑦𝑧) ∈ 𝐿 ∧ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
144, 11, 13syl2anc 587 . . . . . 6 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
15 ineq12 4115 . . . . . . . 8 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (𝑠𝑡) = ((𝐹𝑦) ∩ (𝐹𝑧)))
1615eqeq1d 2761 . . . . . . 7 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ((𝑠𝑡) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1716rexbidv 3222 . . . . . 6 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥) ↔ ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1814, 17syl5ibrcom 250 . . . . 5 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
1918rexlimdvva 3219 . . . 4 (𝜑 → (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
20 imaeq2 5903 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2120eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑦 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑦)))
2221cbvrexvw 3363 . . . . . 6 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑦𝐿 𝑠 = (𝐹𝑦))
23 imaeq2 5903 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2423eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑧 → (𝑡 = (𝐹𝑥) ↔ 𝑡 = (𝐹𝑧)))
2524cbvrexvw 3363 . . . . . 6 (∃𝑥𝐿 𝑡 = (𝐹𝑥) ↔ ∃𝑧𝐿 𝑡 = (𝐹𝑧))
2622, 25anbi12i 629 . . . . 5 ((∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
27 eqid 2759 . . . . . . . 8 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
2827elrnmpt 5803 . . . . . . 7 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
2928elv 3416 . . . . . 6 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
3027elrnmpt 5803 . . . . . . 7 (𝑡 ∈ V → (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
3130elv 3416 . . . . . 6 (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥))
3229, 31anbi12i 629 . . . . 5 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
33 reeanv 3286 . . . . 5 (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
3426, 32, 333bitr4i 306 . . . 4 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)))
35 vex 3414 . . . . . 6 𝑠 ∈ V
3635inex1 5192 . . . . 5 (𝑠𝑡) ∈ V
3727elrnmpt 5803 . . . . 5 ((𝑠𝑡) ∈ V → ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
3836, 37ax-mp 5 . . . 4 ((𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥))
3919, 34, 383imtr4g 299 . . 3 (𝜑 → ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → (𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))))
4039ralrimivv 3120 . 2 (𝜑 → ∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
41 mptexg 6982 . . 3 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
42 rnexg 7621 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
43 inficl 8936 . . 3 (ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
441, 41, 42, 434syl 19 . 2 (𝜑 → (∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥))))
4540, 44mpbid 235 1 (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ∩ cin 3860   ⊆ wss 3861   ↦ cmpt 5117  ◡ccnv 5528  ran crn 5530   “ cima 5532  Fun wfun 6335  ⟶wf 6337  ‘cfv 6341  (class class class)co 7157  ficfi 8921  fBascfbas 20169  Filcfil 22560   FilMap cfm 22648 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-om 7587  df-1o 8119  df-er 8306  df-en 8542  df-fin 8545  df-fi 8922  df-fbas 20178  df-fil 22561 This theorem is referenced by:  fmfnfmlem4  22672
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