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Theorem inelfi 8271
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 4878 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
213adant1 1077 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
3 prfi 8182 . . . . 5 {𝐴, 𝐵} ∈ Fin
43a1i 11 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ Fin)
52, 4elind 3778 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin))
6 intprg 4478 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
763adant1 1077 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
87eqcomd 2627 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) = {𝐴, 𝐵})
9 inteq 4445 . . . . 5 (𝑝 = {𝐴, 𝐵} → 𝑝 = {𝐴, 𝐵})
109eqeq2d 2631 . . . 4 (𝑝 = {𝐴, 𝐵} → ((𝐴𝐵) = 𝑝 ↔ (𝐴𝐵) = {𝐴, 𝐵}))
1110rspcev 3295 . . 3 (({𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐴𝐵) = {𝐴, 𝐵}) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
125, 8, 11syl2anc 692 . 2 ((𝑋𝑉𝐴𝑋𝐵𝑋) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
13 inex1g 4763 . . . 4 (𝐴𝑋 → (𝐴𝐵) ∈ V)
14133ad2ant2 1081 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ V)
15 simp1 1059 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → 𝑋𝑉)
16 elfi 8266 . . 3 (((𝐴𝐵) ∈ V ∧ 𝑋𝑉) → ((𝐴𝐵) ∈ (fi‘𝑋) ↔ ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝))
1714, 15, 16syl2anc 692 . 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 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3555  𝒫 cpw 4132  {cpr 4152   cint 4442  cfv 5849  Fincfn 7902  ficfi 8263
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 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-fin 7906  df-fi 8264
This theorem is referenced by:  neiptoptop  20848  sigapildsyslem  30017
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