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Theorem ackbij1lem7 9251
Description: Lemma for ackbij1 9263. (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 4669 . . 3 (𝑥 = 𝐴 𝑦𝑥 ({𝑦} × 𝒫 𝑦) = 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
21fveq2d 6337 . 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 6343 . 2 (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
52, 3, 4fvmpt 6425 1 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cin 3723  𝒫 cpw 4298  {csn 4317   ciun 4655  cmpt 4864   × cxp 5248  cfv 6032  ωcom 7213  Fincfn 8110  cardccrd 8962
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040
This theorem is referenced by:  ackbij1lem8  9252  ackbij1lem9  9253
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