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Theorem ackbij1lem7 10265
Description: Lemma for ackbij1 10277. (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 5016 . . 3 (𝑥 = 𝐴 𝑦𝑥 ({𝑦} × 𝒫 𝑦) = 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
21fveq2d 6904 . 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 6913 . 2 (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
52, 3, 4fvmpt 7008 1 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cin 3945  𝒫 cpw 4606  {csn 4632   ciun 5000  cmpt 5235   × cxp 5679  cfv 6553  ωcom 7875  Fincfn 8973  cardccrd 9974
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-iota 6505  df-fun 6555  df-fv 6561
This theorem is referenced by:  ackbij1lem8  10266  ackbij1lem9  10267
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