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Theorem elrfirn 40517
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 6607 . . 3 (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2 elrfi 40516 . . 3 ((𝐵𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
31, 2sylan2 593 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
4 imassrn 5980 . . . . . 6 (𝐹𝑣) ⊆ ran 𝐹
5 pwexg 5301 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
6 ssexg 5247 . . . . . . . 8 ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
71, 5, 6syl2anr 597 . . . . . . 7 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8 elpw2g 5268 . . . . . . 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 257 . . . . 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 6603 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
1312ad2antlr 724 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14 inss2 4163 . . . . . . 7 (𝒫 𝐼 ∩ Fin) ⊆ Fin
1514sseli 3917 . . . . . 6 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
1615adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17 imafi 8958 . . . . 5 ((Fun 𝐹𝑣 ∈ Fin) → (𝐹𝑣) ∈ Fin)
1813, 16, 17syl2anc 584 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ Fin)
1911, 18elind 4128 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20 ffn 6600 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵𝐹 Fn 𝐼)
2120ad2antlr 724 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22 inss1 4162 . . . . . . . 8 (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
2322sseli 3917 . . . . . . 7 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
2423elpwid 4544 . . . . . 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 4163 . . . . . . 7 (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
2726sseli 3917 . . . . . 6 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
2827adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29 fipreima 9125 . . . . 5 ((𝐹 Fn 𝐼𝑤 ⊆ ran 𝐹𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
3021, 25, 28, 29syl3anc 1370 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
31 eqcom 2745 . . . . 5 ((𝐹𝑣) = 𝑤𝑤 = (𝐹𝑣))
3231rexbii 3181 . . . 4 (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
3330, 32sylib 217 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
34 inteq 4882 . . . . . 6 (𝑤 = (𝐹𝑣) → 𝑤 = (𝐹𝑣))
3534ineq2d 4146 . . . . 5 (𝑤 = (𝐹𝑣) → (𝐵 𝑤) = (𝐵 (𝐹𝑣)))
3635eqeq2d 2749 . . . 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 5332 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣))))
3920ad2antlr 724 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40 inss1 4162 . . . . . . . . . 10 (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
4140sseli 3917 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
4241elpwid 4544 . . . . . . . 8 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣𝐼)
4342adantl 482 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣𝐼)
44 imaiinfv 40515 . . . . . . 7 ((𝐹 Fn 𝐼𝑣𝐼) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4539, 43, 44syl2anc 584 . . . . . 6 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4645eqcomd 2744 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) = 𝑦𝑣 (𝐹𝑦))
4746ineq2d 4146 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 (𝐹𝑣)) = (𝐵 𝑦𝑣 (𝐹𝑦)))
4847eqeq2d 2749 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 (𝐹𝑣)) ↔ 𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
4948rexbidva 3225 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
503, 38, 493bitrd 305 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 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  cun 3885  cin 3886  wss 3887  𝒫 cpw 4533  {csn 4561   cint 4879   ciin 4925  ran crn 5590  cima 5592  Fun wfun 6427   Fn wfn 6428  wf 6429  cfv 6433  Fincfn 8733  ficfi 9169
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737  df-fi 9170
This theorem is referenced by:  elrfirn2  40518
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