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Theorem inelfi 8491
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 5064 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
213adant1 1125 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ 𝒫 𝑋)
3 prfi 8402 . . . . 5 {𝐴, 𝐵} ∈ Fin
43a1i 11 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ Fin)
52, 4elind 3941 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin))
6 intprg 4663 . . . . 5 ((𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
763adant1 1125 . . . 4 ((𝑋𝑉𝐴𝑋𝐵𝑋) → {𝐴, 𝐵} = (𝐴𝐵))
87eqcomd 2766 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) = {𝐴, 𝐵})
9 inteq 4630 . . . . 5 (𝑝 = {𝐴, 𝐵} → 𝑝 = {𝐴, 𝐵})
109eqeq2d 2770 . . . 4 (𝑝 = {𝐴, 𝐵} → ((𝐴𝐵) = 𝑝 ↔ (𝐴𝐵) = {𝐴, 𝐵}))
1110rspcev 3449 . . 3 (({𝐴, 𝐵} ∈ (𝒫 𝑋 ∩ Fin) ∧ (𝐴𝐵) = {𝐴, 𝐵}) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
125, 8, 11syl2anc 696 . 2 ((𝑋𝑉𝐴𝑋𝐵𝑋) → ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝)
13 inex1g 4953 . . . 4 (𝐴𝑋 → (𝐴𝐵) ∈ V)
14133ad2ant2 1129 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝐴𝐵) ∈ V)
15 simp1 1131 . . 3 ((𝑋𝑉𝐴𝑋𝐵𝑋) → 𝑋𝑉)
16 elfi 8486 . . 3 (((𝐴𝐵) ∈ V ∧ 𝑋𝑉) → ((𝐴𝐵) ∈ (fi‘𝑋) ↔ ∃𝑝 ∈ (𝒫 𝑋 ∩ Fin)(𝐴𝐵) = 𝑝))
1714, 15, 16syl2anc 696 . 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 1072   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  cin 3714  𝒫 cpw 4302  {cpr 4323   cint 4627  cfv 6049  Fincfn 8123  ficfi 8483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-fin 8127  df-fi 8484
This theorem is referenced by:  neiptoptop  21157  sigapildsyslem  30554
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