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Theorem elrfirn 36765
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 6015 . . 3 (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2 elrfi 36764 . . 3 ((𝐵𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
31, 2sylan2 491 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤)))
4 imassrn 5441 . . . . . 6 (𝐹𝑣) ⊆ ran 𝐹
5 pwexg 4815 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
6 ssexg 4769 . . . . . . . 8 ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
71, 5, 6syl2anr 495 . . . . . . 7 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8 elpw2g 4792 . . . . . . 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 248 . . . . 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 6010 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
1312ad2antlr 762 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14 inss2 3817 . . . . . . 7 (𝒫 𝐼 ∩ Fin) ⊆ Fin
1514sseli 3583 . . . . . 6 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
1615adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17 imafi 8210 . . . . 5 ((Fun 𝐹𝑣 ∈ Fin) → (𝐹𝑣) ∈ Fin)
1813, 16, 17syl2anc 692 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ Fin)
1911, 18elind 3781 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20 ffn 6007 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵𝐹 Fn 𝐼)
2120ad2antlr 762 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22 inss1 3816 . . . . . . . 8 (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
2322sseli 3583 . . . . . . 7 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
2423elpwid 4146 . . . . . 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 3817 . . . . . . 7 (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
2726sseli 3583 . . . . . 6 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
2827adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29 fipreima 8223 . . . . 5 ((𝐹 Fn 𝐼𝑤 ⊆ ran 𝐹𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
3021, 25, 28, 29syl3anc 1323 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
31 eqcom 2628 . . . . 5 ((𝐹𝑣) = 𝑤𝑤 = (𝐹𝑣))
3231rexbii 3035 . . . 4 (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
3330, 32sylib 208 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹𝑣))
34 inteq 4448 . . . . . 6 (𝑤 = (𝐹𝑣) → 𝑤 = (𝐹𝑣))
3534ineq2d 3797 . . . . 5 (𝑤 = (𝐹𝑣) → (𝐵 𝑤) = (𝐵 (𝐹𝑣)))
3635eqeq2d 2631 . . . 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 4846 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣))))
3920ad2antlr 762 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40 inss1 3816 . . . . . . . . . 10 (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
4140sseli 3583 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
4241elpwid 4146 . . . . . . . 8 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣𝐼)
4342adantl 482 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣𝐼)
44 imaiinfv 36763 . . . . . . 7 ((𝐹 Fn 𝐼𝑣𝐼) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4539, 43, 44syl2anc 692 . . . . . 6 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4645eqcomd 2627 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) = 𝑦𝑣 (𝐹𝑦))
4746ineq2d 3797 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 (𝐹𝑣)) = (𝐵 𝑦𝑣 (𝐹𝑦)))
4847eqeq2d 2631 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 (𝐹𝑣)) ↔ 𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
4948rexbidva 3043 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
503, 38, 493bitrd 294 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 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3189  cun 3557  cin 3558  wss 3559  𝒫 cpw 4135  {csn 4153   cint 4445   ciin 4491  ran crn 5080  cima 5082  Fun wfun 5846   Fn wfn 5847  wf 5848  cfv 5852  Fincfn 7906  ficfi 8267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-fin 7910  df-fi 8268
This theorem is referenced by:  elrfirn2  36766
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