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Theorem heiborlem2 37136
Description: Lemma for heibor 37145. Substitutions for the set 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpenβ€˜π·)
heibor.3 𝐾 = {𝑒 ∣ Β¬ βˆƒπ‘£ ∈ (𝒫 π‘ˆ ∩ Fin)𝑒 βŠ† βˆͺ 𝑣}
heibor.4 𝐺 = {βŸ¨π‘¦, π‘›βŸ© ∣ (𝑛 ∈ β„•0 ∧ 𝑦 ∈ (πΉβ€˜π‘›) ∧ (𝑦𝐡𝑛) ∈ 𝐾)}
heiborlem2.5 𝐴 ∈ V
heiborlem2.6 𝐢 ∈ V
Assertion
Ref Expression
heiborlem2 (𝐴𝐺𝐢 ↔ (𝐢 ∈ β„•0 ∧ 𝐴 ∈ (πΉβ€˜πΆ) ∧ (𝐴𝐡𝐢) ∈ 𝐾))
Distinct variable groups:   𝑦,𝑛,𝐴   𝑒,𝑛,𝐹,𝑦   𝑣,𝑛,𝐷,𝑒,𝑦   𝐡,𝑛,𝑒,𝑣,𝑦   𝑛,𝐽,𝑒,𝑣,𝑦   π‘ˆ,𝑛,𝑒,𝑣,𝑦   𝐢,𝑛,𝑒,𝑣,𝑦   𝑛,𝐾,𝑦
Allowed substitution hints:   𝐴(𝑣,𝑒)   𝐹(𝑣)   𝐺(𝑦,𝑣,𝑒,𝑛)   𝐾(𝑣,𝑒)

Proof of Theorem heiborlem2
StepHypRef Expression
1 heiborlem2.5 . 2 𝐴 ∈ V
2 heiborlem2.6 . 2 𝐢 ∈ V
3 eleq1 2813 . . 3 (𝑦 = 𝐴 β†’ (𝑦 ∈ (πΉβ€˜π‘›) ↔ 𝐴 ∈ (πΉβ€˜π‘›)))
4 oveq1 7408 . . . 4 (𝑦 = 𝐴 β†’ (𝑦𝐡𝑛) = (𝐴𝐡𝑛))
54eleq1d 2810 . . 3 (𝑦 = 𝐴 β†’ ((𝑦𝐡𝑛) ∈ 𝐾 ↔ (𝐴𝐡𝑛) ∈ 𝐾))
63, 53anbi23d 1435 . 2 (𝑦 = 𝐴 β†’ ((𝑛 ∈ β„•0 ∧ 𝑦 ∈ (πΉβ€˜π‘›) ∧ (𝑦𝐡𝑛) ∈ 𝐾) ↔ (𝑛 ∈ β„•0 ∧ 𝐴 ∈ (πΉβ€˜π‘›) ∧ (𝐴𝐡𝑛) ∈ 𝐾)))
7 eleq1 2813 . . 3 (𝑛 = 𝐢 β†’ (𝑛 ∈ β„•0 ↔ 𝐢 ∈ β„•0))
8 fveq2 6881 . . . 4 (𝑛 = 𝐢 β†’ (πΉβ€˜π‘›) = (πΉβ€˜πΆ))
98eleq2d 2811 . . 3 (𝑛 = 𝐢 β†’ (𝐴 ∈ (πΉβ€˜π‘›) ↔ 𝐴 ∈ (πΉβ€˜πΆ)))
10 oveq2 7409 . . . 4 (𝑛 = 𝐢 β†’ (𝐴𝐡𝑛) = (𝐴𝐡𝐢))
1110eleq1d 2810 . . 3 (𝑛 = 𝐢 β†’ ((𝐴𝐡𝑛) ∈ 𝐾 ↔ (𝐴𝐡𝐢) ∈ 𝐾))
127, 9, 113anbi123d 1432 . 2 (𝑛 = 𝐢 β†’ ((𝑛 ∈ β„•0 ∧ 𝐴 ∈ (πΉβ€˜π‘›) ∧ (𝐴𝐡𝑛) ∈ 𝐾) ↔ (𝐢 ∈ β„•0 ∧ 𝐴 ∈ (πΉβ€˜πΆ) ∧ (𝐴𝐡𝐢) ∈ 𝐾)))
13 heibor.4 . 2 𝐺 = {βŸ¨π‘¦, π‘›βŸ© ∣ (𝑛 ∈ β„•0 ∧ 𝑦 ∈ (πΉβ€˜π‘›) ∧ (𝑦𝐡𝑛) ∈ 𝐾)}
141, 2, 6, 12, 13brab 5533 1 (𝐴𝐺𝐢 ↔ (𝐢 ∈ β„•0 ∧ 𝐴 ∈ (πΉβ€˜πΆ) ∧ (𝐴𝐡𝐢) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2701  βˆƒwrex 3062  Vcvv 3466   ∩ cin 3939   βŠ† wss 3940  π’« cpw 4594  βˆͺ cuni 4899   class class class wbr 5138  {copab 5200  β€˜cfv 6533  (class class class)co 7401  Fincfn 8934  β„•0cn0 12468  MetOpencmopn 21217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-iota 6485  df-fv 6541  df-ov 7404
This theorem is referenced by:  heiborlem3  37137  heiborlem5  37139  heiborlem6  37140  heiborlem8  37142  heiborlem10  37144
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