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Theorem fiin 8884
Description: The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
fiin ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴𝐵) ∈ (fi‘𝐶))

Proof of Theorem fiin
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6695 . . . . . 6 (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2 elfi 8875 . . . . . 6 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥))
31, 2mpdan 686 . . . . 5 (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥))
43ibi 270 . . . 4 (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥)
54adantr 484 . . 3 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥)
6 simpr 488 . . . 4 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7 elfi 8875 . . . . . 6 ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
87ancoms 462 . . . . 5 ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
91, 8sylan 583 . . . 4 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
106, 9mpbid 235 . . 3 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦)
11 elin 3936 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin))
12 elin 3936 . . . . . . . . 9 (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin))
13 pwuncl 7487 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝑦) ∈ 𝒫 𝐶)
14 unfi 8783 . . . . . . . . . . 11 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
1513, 14anim12i 615 . . . . . . . . . 10 (((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
1615an4s 659 . . . . . . . . 9 (((𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
1711, 12, 16syl2anb 600 . . . . . . . 8 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
18 elin 3936 . . . . . . . 8 ((𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
1917, 18sylibr 237 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin))
20 ineq12 4170 . . . . . . . 8 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = ( 𝑥 𝑦))
21 intun 4895 . . . . . . . 8 (𝑥𝑦) = ( 𝑥 𝑦)
2220, 21syl6eqr 2877 . . . . . . 7 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = (𝑥𝑦))
23 inteq 4866 . . . . . . . 8 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
2423rspceeqv 3625 . . . . . . 7 (((𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴𝐵) = (𝑥𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
2519, 22, 24syl2an 598 . . . . . 6 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = 𝑥𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
2625an4s 659 . . . . 5 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
2726rexlimdvaa 3278 . . . 4 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
2827rexlimiva 3274 . . 3 (∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥 → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
295, 10, 28sylc 65 . 2 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
30 inex1g 5210 . . . 4 (𝐴 ∈ (fi‘𝐶) → (𝐴𝐵) ∈ V)
31 elfi 8875 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
3230, 1, 31syl2anc 587 . . 3 (𝐴 ∈ (fi‘𝐶) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
3332adantr 484 . 2 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
3429, 33mpbird 260 1 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴𝐵) ∈ (fi‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wrex 3134  Vcvv 3481  cun 3918  cin 3919  𝒫 cpw 4523   cint 4863  cfv 6344  Fincfn 8506  ficfi 8872
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-oadd 8103  df-er 8286  df-en 8507  df-fin 8510  df-fi 8873
This theorem is referenced by:  dffi2  8885  inficl  8887  elfiun  8892  dffi3  8893  fibas  21588  ordtbas2  21802  fsubbas  22478
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