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Theorem fiin 8272
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 6178 . . . . . 6 (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2 elfi 8263 . . . . . 6 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥))
31, 2mpdan 701 . . . . 5 (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥))
43ibi 256 . . . 4 (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥)
54adantr 481 . . 3 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥)
6 simpr 477 . . . 4 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7 elfi 8263 . . . . . 6 ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
87ancoms 469 . . . . 5 ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
91, 8sylan 488 . . . 4 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
106, 9mpbid 222 . . 3 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦)
11 elin 3774 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin))
12 elin 3774 . . . . . . . . 9 (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin))
13 elpwi 4140 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝐶𝑥𝐶)
14 elpwi 4140 . . . . . . . . . . . . . 14 (𝑦 ∈ 𝒫 𝐶𝑦𝐶)
1513, 14anim12i 589 . . . . . . . . . . . . 13 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝐶𝑦𝐶))
16 unss 3765 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦𝐶) ↔ (𝑥𝑦) ⊆ 𝐶)
1715, 16sylib 208 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝑦) ⊆ 𝐶)
18 vex 3189 . . . . . . . . . . . . . 14 𝑥 ∈ V
19 vex 3189 . . . . . . . . . . . . . 14 𝑦 ∈ V
2018, 19unex 6909 . . . . . . . . . . . . 13 (𝑥𝑦) ∈ V
2120elpw 4136 . . . . . . . . . . . 12 ((𝑥𝑦) ∈ 𝒫 𝐶 ↔ (𝑥𝑦) ⊆ 𝐶)
2217, 21sylibr 224 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝑦) ∈ 𝒫 𝐶)
23 unfi 8171 . . . . . . . . . . 11 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
2422, 23anim12i 589 . . . . . . . . . 10 (((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
2524an4s 868 . . . . . . . . 9 (((𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
2611, 12, 25syl2anb 496 . . . . . . . 8 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
27 elin 3774 . . . . . . . 8 ((𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
2826, 27sylibr 224 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin))
29 ineq12 3787 . . . . . . . 8 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = ( 𝑥 𝑦))
30 intun 4474 . . . . . . . 8 (𝑥𝑦) = ( 𝑥 𝑦)
3129, 30syl6eqr 2673 . . . . . . 7 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = (𝑥𝑦))
32 inteq 4443 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
3332eqeq2d 2631 . . . . . . . 8 (𝑧 = (𝑥𝑦) → ((𝐴𝐵) = 𝑧 ↔ (𝐴𝐵) = (𝑥𝑦)))
3433rspcev 3295 . . . . . . 7 (((𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴𝐵) = (𝑥𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3528, 31, 34syl2an 494 . . . . . 6 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = 𝑥𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3635an4s 868 . . . . 5 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3736rexlimdvaa 3025 . . . 4 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
3837rexlimiva 3021 . . 3 (∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥 → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
395, 10, 38sylc 65 . 2 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
40 inex1g 4761 . . . 4 (𝐴 ∈ (fi‘𝐶) → (𝐴𝐵) ∈ V)
41 elfi 8263 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
4240, 1, 41syl2anc 692 . . 3 (𝐴 ∈ (fi‘𝐶) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
4342adantr 481 . 2 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
4439, 43mpbird 247 1 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴𝐵) ∈ (fi‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130   cint 4440  cfv 5847  Fincfn 7899  ficfi 8260
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261
This theorem is referenced by:  dffi2  8273  inficl  8275  elfiun  8280  dffi3  8281  fibas  20692  ordtbas2  20905  fsubbas  21581
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