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Theorem fmfnfmlem3 22480
 Description: Lemma for fmfnfm 22482. (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 22378 . . . . . . . . 9 ((𝐿 ∈ (Fil‘𝑋) ∧ 𝑦𝐿𝑧𝐿) → (𝑦𝑧) ∈ 𝐿)
323expb 1114 . . . . . . . 8 ((𝐿 ∈ (Fil‘𝑋) ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
41, 3sylan 580 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → (𝑦𝑧) ∈ 𝐿)
5 fmfnfm.f . . . . . . . . 9 (𝜑𝐹:𝑌𝑋)
6 ffun 6514 . . . . . . . . 9 (𝐹:𝑌𝑋 → Fun 𝐹)
7 funcnvcnv 6418 . . . . . . . . 9 (Fun 𝐹 → Fun 𝐹)
8 imain 6436 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑦𝑧)) = ((𝐹𝑦) ∩ (𝐹𝑧)))
98eqcomd 2832 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
105, 6, 7, 94syl 19 . . . . . . . 8 (𝜑 → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
1110adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧)))
12 imaeq2 5923 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝐹𝑥) = (𝐹 “ (𝑦𝑧)))
1312rspceeqv 3642 . . . . . . 7 (((𝑦𝑧) ∈ 𝐿 ∧ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹 “ (𝑦𝑧))) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
144, 11, 13syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥))
15 ineq12 4188 . . . . . . . 8 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (𝑠𝑡) = ((𝐹𝑦) ∩ (𝐹𝑧)))
1615eqeq1d 2828 . . . . . . 7 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ((𝑠𝑡) = (𝐹𝑥) ↔ ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1716rexbidv 3302 . . . . . 6 ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → (∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥) ↔ ∃𝑥𝐿 ((𝐹𝑦) ∩ (𝐹𝑧)) = (𝐹𝑥)))
1814, 17syl5ibrcom 248 . . . . 5 ((𝜑 ∧ (𝑦𝐿𝑧𝐿)) → ((𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
1918rexlimdvva 3299 . . . 4 (𝜑 → (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) → ∃𝑥𝐿 (𝑠𝑡) = (𝐹𝑥)))
20 imaeq2 5923 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2120eqeq2d 2837 . . . . . . 7 (𝑥 = 𝑦 → (𝑠 = (𝐹𝑥) ↔ 𝑠 = (𝐹𝑦)))
2221cbvrexvw 3456 . . . . . 6 (∃𝑥𝐿 𝑠 = (𝐹𝑥) ↔ ∃𝑦𝐿 𝑠 = (𝐹𝑦))
23 imaeq2 5923 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2423eqeq2d 2837 . . . . . . 7 (𝑥 = 𝑧 → (𝑡 = (𝐹𝑥) ↔ 𝑡 = (𝐹𝑧)))
2524cbvrexvw 3456 . . . . . 6 (∃𝑥𝐿 𝑡 = (𝐹𝑥) ↔ ∃𝑧𝐿 𝑡 = (𝐹𝑧))
2622, 25anbi12i 626 . . . . 5 ((∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
27 eqid 2826 . . . . . . . 8 (𝑥𝐿 ↦ (𝐹𝑥)) = (𝑥𝐿 ↦ (𝐹𝑥))
2827elrnmpt 5827 . . . . . . 7 (𝑠 ∈ V → (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥)))
2928elv 3505 . . . . . 6 (𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑠 = (𝐹𝑥))
3027elrnmpt 5827 . . . . . . 7 (𝑡 ∈ V → (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
3130elv 3505 . . . . . 6 (𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ↔ ∃𝑥𝐿 𝑡 = (𝐹𝑥))
3229, 31anbi12i 626 . . . . 5 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ (∃𝑥𝐿 𝑠 = (𝐹𝑥) ∧ ∃𝑥𝐿 𝑡 = (𝐹𝑥)))
33 reeanv 3373 . . . . 5 (∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)) ↔ (∃𝑦𝐿 𝑠 = (𝐹𝑦) ∧ ∃𝑧𝐿 𝑡 = (𝐹𝑧)))
3426, 32, 333bitr4i 304 . . . 4 ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) ↔ ∃𝑦𝐿𝑧𝐿 (𝑠 = (𝐹𝑦) ∧ 𝑡 = (𝐹𝑧)))
35 vex 3503 . . . . . 6 𝑠 ∈ V
3635inex1 5218 . . . . 5 (𝑠𝑡) ∈ V
3727elrnmpt 5827 . . . . 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 297 . . 3 (𝜑 → ((𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)) ∧ 𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))) → (𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))))
4039ralrimivv 3195 . 2 (𝜑 → ∀𝑠 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))∀𝑡 ∈ ran (𝑥𝐿 ↦ (𝐹𝑥))(𝑠𝑡) ∈ ran (𝑥𝐿 ↦ (𝐹𝑥)))
41 mptexg 6979 . . 3 (𝐿 ∈ (Fil‘𝑋) → (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
42 rnexg 7602 . . 3 ((𝑥𝐿 ↦ (𝐹𝑥)) ∈ V → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ V)
43 inficl 8878 . . 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 233 1 (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∩ cin 3939   ⊆ wss 3940   ↦ cmpt 5143  ◡ccnv 5553  ran crn 5555   “ cima 5557  Fun wfun 6346  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148  ficfi 8863  fBascfbas 20449  Filcfil 22369   FilMap cfm 22457 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-fin 8502  df-fi 8864  df-fbas 20458  df-fil 22370 This theorem is referenced by:  fmfnfmlem4  22481
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