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Theorem ackbij1lem7 8931
 Description: Lemma for ackbij1 8943. (Contributed by Stefan O'Rear, 21-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
Assertion
Ref Expression
ackbij1lem7 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij1lem7
StepHypRef Expression
1 iuneq1 4470 . . 3 (𝑥 = 𝐴 𝑦𝑥 ({𝑦} × 𝒫 𝑦) = 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
21fveq2d 6107 . 2 (𝑥 = 𝐴 → (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
3 ackbij.f . 2 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
4 fvex 6113 . 2 (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
52, 3, 4fvmpt 6191 1 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  𝒫 cpw 4108  {csn 4125  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  ωcom 6957  Fincfn 7841  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  ackbij1lem8  8932  ackbij1lem9  8933
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