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Theorem fiin 8484
 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 6362 . . . . . 6 (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2 elfi 8475 . . . . . 6 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥))
31, 2mpdan 667 . . . . 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 466 . . 3 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = 𝑥)
6 simpr 471 . . . 4 ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7 elfi 8475 . . . . . 6 ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
87ancoms 455 . . . . 5 ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦))
91, 8sylan 569 . . . 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 3947 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin))
12 elin 3947 . . . . . . . . 9 (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin))
13 elpwi 4307 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝒫 𝐶𝑥𝐶)
14 elpwi 4307 . . . . . . . . . . . . . 14 (𝑦 ∈ 𝒫 𝐶𝑦𝐶)
1513, 14anim12i 600 . . . . . . . . . . . . 13 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝐶𝑦𝐶))
16 unss 3938 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦𝐶) ↔ (𝑥𝑦) ⊆ 𝐶)
1715, 16sylib 208 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝑦) ⊆ 𝐶)
18 vex 3354 . . . . . . . . . . . . . 14 𝑥 ∈ V
19 vex 3354 . . . . . . . . . . . . . 14 𝑦 ∈ V
2018, 19unex 7103 . . . . . . . . . . . . 13 (𝑥𝑦) ∈ V
2120elpw 4303 . . . . . . . . . . . 12 ((𝑥𝑦) ∈ 𝒫 𝐶 ↔ (𝑥𝑦) ⊆ 𝐶)
2217, 21sylibr 224 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) → (𝑥𝑦) ∈ 𝒫 𝐶)
23 unfi 8383 . . . . . . . . . . 11 ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥𝑦) ∈ Fin)
2422, 23anim12i 600 . . . . . . . . . 10 (((𝑥 ∈ 𝒫 𝐶𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
2524an4s 639 . . . . . . . . 9 (((𝑥 ∈ 𝒫 𝐶𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶𝑦 ∈ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
2611, 12, 25syl2anb 585 . . . . . . . 8 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥𝑦) ∈ 𝒫 𝐶 ∧ (𝑥𝑦) ∈ Fin))
27 elin 3947 . . . . . . . 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 3960 . . . . . . . 8 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = ( 𝑥 𝑦))
30 intun 4643 . . . . . . . 8 (𝑥𝑦) = ( 𝑥 𝑦)
3129, 30syl6eqr 2823 . . . . . . 7 ((𝐴 = 𝑥𝐵 = 𝑦) → (𝐴𝐵) = (𝑥𝑦))
32 inteq 4614 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → 𝑧 = (𝑥𝑦))
3332eqeq2d 2781 . . . . . . . 8 (𝑧 = (𝑥𝑦) → ((𝐴𝐵) = 𝑧 ↔ (𝐴𝐵) = (𝑥𝑦)))
3433rspcev 3460 . . . . . . 7 (((𝑥𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴𝐵) = (𝑥𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3528, 31, 34syl2an 583 . . . . . 6 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = 𝑥𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3635an4s 639 . . . . 5 (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧)
3736rexlimdvaa 3180 . . . 4 ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
3837rexlimiva 3176 . . 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 4935 . . . 4 (𝐴 ∈ (fi‘𝐶) → (𝐴𝐵) ∈ V)
41 elfi 8475 . . . 4 (((𝐴𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
4240, 1, 41syl2anc 573 . . 3 (𝐴 ∈ (fi‘𝐶) → ((𝐴𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴𝐵) = 𝑧))
4342adantr 466 . 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 382   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  Vcvv 3351   ∪ cun 3721   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4297  ∩ 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-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473 This theorem is referenced by:  dffi2  8485  inficl  8487  elfiun  8492  dffi3  8493  fibas  21002  ordtbas2  21216  fsubbas  21891
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