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Theorem elrfirn 41004
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 6675 . . 3 (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2 elrfi 41003 . . 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 6024 . . . . . 6 (𝐹𝑣) ⊆ ran 𝐹
5 pwexg 5333 . . . . . . . 8 (𝐵𝑉 → 𝒫 𝐵 ∈ V)
6 ssexg 5280 . . . . . . . 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 5301 . . . . . . 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 6671 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
1312ad2antlr 725 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14 inss2 4189 . . . . . . 7 (𝒫 𝐼 ∩ Fin) ⊆ Fin
1514sseli 3940 . . . . . 6 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
1615adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17 imafi 9119 . . . . 5 ((Fun 𝐹𝑣 ∈ Fin) → (𝐹𝑣) ∈ Fin)
1813, 16, 17syl2anc 584 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ Fin)
1911, 18elind 4154 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20 ffn 6668 . . . . . 6 (𝐹:𝐼⟶𝒫 𝐵𝐹 Fn 𝐼)
2120ad2antlr 725 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22 inss1 4188 . . . . . . . 8 (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
2322sseli 3940 . . . . . . 7 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
2423elpwid 4569 . . . . . 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 4189 . . . . . . 7 (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
2726sseli 3940 . . . . . 6 (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
2827adantl 482 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29 fipreima 9302 . . . . 5 ((𝐹 Fn 𝐼𝑤 ⊆ ran 𝐹𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
3021, 25, 28, 29syl3anc 1371 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹𝑣) = 𝑤)
31 eqcom 2743 . . . . 5 ((𝐹𝑣) = 𝑤𝑤 = (𝐹𝑣))
3231rexbii 3097 . . . 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 4910 . . . . . 6 (𝑤 = (𝐹𝑣) → 𝑤 = (𝐹𝑣))
3534ineq2d 4172 . . . . 5 (𝑤 = (𝐹𝑣) → (𝐵 𝑤) = (𝐵 (𝐹𝑣)))
3635eqeq2d 2747 . . . 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 5364 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 𝑤) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣))))
3920ad2antlr 725 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40 inss1 4188 . . . . . . . . . 10 (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
4140sseli 3940 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
4241elpwid 4569 . . . . . . . 8 (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣𝐼)
4342adantl 482 . . . . . . 7 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣𝐼)
44 imaiinfv 41002 . . . . . . 7 ((𝐹 Fn 𝐼𝑣𝐼) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4539, 43, 44syl2anc 584 . . . . . 6 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑦𝑣 (𝐹𝑦) = (𝐹𝑣))
4645eqcomd 2742 . . . . 5 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹𝑣) = 𝑦𝑣 (𝐹𝑦))
4746ineq2d 4172 . . . 4 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 (𝐹𝑣)) = (𝐵 𝑦𝑣 (𝐹𝑦)))
4847eqeq2d 2747 . . 3 (((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 (𝐹𝑣)) ↔ 𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
4948rexbidva 3173 . 2 ((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 (𝐹𝑣)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))
503, 38, 493bitrd 304 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 1541  wcel 2106  wrex 3073  Vcvv 3445  cun 3908  cin 3909  wss 3910  𝒫 cpw 4560  {csn 4586   cint 4907   ciin 4955  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  Fincfn 8883  ficfi 9346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-en 8884  df-fin 8887  df-fi 9347
This theorem is referenced by:  elrfirn2  41005
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