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Theorem elrfirn 39170
Description: Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
elrfirn ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐵   𝑣,𝐹,𝑦   𝑣,𝐼   𝑣,𝑉   𝑦,𝑣
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐼(𝑦)   𝑉(𝑦)

Proof of Theorem elrfirn
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 frn 6513 . . 3 (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2 elrfi 39169 . . 3 ((𝐵𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
31, 2sylan2 592 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
4 imassrn 5933 . . . . . 6 (𝐹𝑣) ⊆ ran 𝐹
5 pwexg 5270 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
6 ssexg 5218 . . . . . . . 8 ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
71, 5, 6syl2anr 596 . . . . . . 7 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8 elpw2g 5238 . . . . . . 7 (ran 𝐹 ∈ V → ((𝐹𝑣) ∈ 𝒫 ran 𝐹 ↔ (𝐹𝑣) ⊆ ran 𝐹))
97, 8syl 17 . . . . . 6 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → ((𝐹𝑣) ∈ 𝒫 ran 𝐹 ↔ (𝐹𝑣) ⊆ ran 𝐹))
104, 9mpbiri 259 . . . . 5 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐹𝑣) ∈ 𝒫 ran 𝐹)
1110adantr 481 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ 𝒫 ran 𝐹)
12 ffun 6510 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
1312ad2antlr 723 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14 inss2 4203 . . . . . . 7 (𝒫 𝐼 ∩ Fin) ⊆ Fin
1514sseli 3960 . . . . . 6 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
1615adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17 imafi 8805 . . . . 5 ((Fun 𝐹𝑣 ∈ Fin) → (𝐹𝑣) ∈ Fin)
1813, 16, 17syl2anc 584 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ Fin)
1911, 18elind 4168 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20 ffn 6507 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵𝐹 Fn 𝐼)
2120ad2antlr 723 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22 inss1 4202 . . . . . . . 8 (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
2322sseli 3960 . . . . . . 7 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
2423elpwid 4549 . . . . . 6 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ⊆ ran 𝐹)
2524adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ⊆ ran 𝐹)
26 inss2 4203 . . . . . . 7 (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
2726sseli 3960 . . . . . 6 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
2827adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29 fipreima 8818 . . . . 5 ((𝐹 Fn 𝐼𝑤 ⊆ ran 𝐹𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
3021, 25, 28, 29syl3anc 1363 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
31 eqcom 2825 . . . . 5 ((𝐹𝑣) = 𝑤𝑤 = (𝐹𝑣))
3231rexbii 3244 . . . 4 (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
3330, 32sylib 219 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
34 inteq 4870 . . . . . 6 (𝑤 = (𝐹𝑣) → 𝑤 = (𝐹𝑣))
3534ineq2d 4186 . . . . 5 (𝑤 = (𝐹𝑣) → (𝐵 𝑤) = (𝐵 (𝐹𝑣)))
3635eqeq2d 2829 . . . 4 (𝑤 = (𝐹𝑣) → (𝐴 = (𝐵 𝑤) ↔ 𝐴 = (𝐵 (𝐹𝑣))))
3736adantl 482 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 = (𝐹𝑣)) → (𝐴 = (𝐵 𝑤) ↔ 𝐴 = (𝐵 (𝐹𝑣))))
3819, 33, 37rexxfrd 5300 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣))))
3920ad2antlr 723 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40 inss1 4202 . . . . . . . . . 10 (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
4140sseli 3960 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
4241elpwid 4549 . . . . . . . 8 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣𝐼)
4342adantl 482 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣𝐼)
44 imaiinfv 39168 . . . . . . 7 ((𝐹 Fn 𝐼𝑣𝐼) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4539, 43, 44syl2anc 584 . . . . . 6 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4645eqcomd 2824 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) = 𝑦𝑣 (𝐹𝑦))
4746ineq2d 4186 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 (𝐹𝑣)) = (𝐵 𝑦𝑣 (𝐹𝑦)))
4847eqeq2d 2829 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 (𝐹𝑣)) ↔ 𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
4948rexbidva 3293 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
503, 38, 493bitrd 306 1 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  cun 3931  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557   cint 4867   ciin 4911  ran crn 5549  cima 5551  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  Fincfn 8497  ficfi 8862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863
This theorem is referenced by:  elrfirn2  39171
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