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Theorem inelfi 8480
Description: The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
Assertion
Ref Expression
inelfi ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ (fi‘𝑋))

Proof of Theorem inelfi
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 prelpwi 5043 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
213adant1 1124 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
3 prfi 8391 . . . . 5 {𝐴, 𝐵} ∈ Fin
43a1i 11 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ Fin)
52, 4elind 3949 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin))
6 intprg 4645 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
763adant1 1124 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
87eqcomd 2777 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) = {𝐴, 𝐵})
9 inteq 4614 . . . . 5 (𝑝 = {𝐴, 𝐵} → 𝑝 = {𝐴, 𝐵})
109eqeq2d 2781 . . . 4 (𝑝 = {𝐴, 𝐵} → ((𝐴𝐵) = 𝑝 ↔ (𝐴𝐵) = {𝐴, 𝐵}))
1110rspcev 3460 . . 3 (({𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐴𝐵) = {𝐴, 𝐵}) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
125, 8, 11syl2anc 573 . 2 ((𝑋𝑉𝐴𝑋𝐵𝑋) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
13 inex1g 4935 . . . 4 (𝐴𝑋 → (𝐴𝐵) ∈ V)
14133ad2ant2 1128 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ V)
15 simp1 1130 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → 𝑋𝑉)
16 elfi 8475 . . 3 (((𝐴𝐵) ∈ V ∧ 𝑋𝑉) → ((𝐴𝐵) ∈ (fi‘𝑋) ↔ ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝))
1714, 15, 16syl2anc 573 . 2 ((𝑋𝑉𝐴𝑋𝐵𝑋) → ((𝐴𝐵) ∈ (fi‘𝑋) ↔ ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝))
1812, 17mpbird 247 1 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ (fi‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  cin 3722  𝒫 cpw 4297  {cpr 4318   cint 4611  cfv 6031  Fincfn 8109  ficfi 8472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473
This theorem is referenced by:  neiptoptop  21156  sigapildsyslem  30564
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