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Theorem ackbij1lem7 9975
Description: Lemma for ackbij1 9987. (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 4946 . . 3 (𝑥 = 𝐴 𝑦𝑥 ({𝑦} × 𝒫 𝑦) = 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
21fveq2d 6773 . 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 6782 . 2 (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
52, 3, 4fvmpt 6870 1 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  cin 3891  𝒫 cpw 4539  {csn 4567   ciun 4930  cmpt 5162   × cxp 5587  cfv 6431  ωcom 7701  Fincfn 8708  cardccrd 9686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6389  df-fun 6433  df-fv 6439
This theorem is referenced by:  ackbij1lem8  9976  ackbij1lem9  9977
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