Theorem ackbij1lem7 10138

Description: Lemma for ackbij1 10150. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem7	⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij1lem7

Step	Hyp	Ref	Expression
1		iuneq1 4938	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
2	1	fveq2d 6831	. 2 ⊢ (𝑥 = 𝐴 → (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
3		ackbij.f	. 2 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
4		fvex 6840	. 2 ⊢ (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
5	2, 3, 4	fvmpt 6935	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∩ cin 3882  𝒫 cpw 4529  {csn 4555  ∪ ciun 4921   ↦ cmpt 5153   × cxp 5616  ‘cfv 6485  ωcom 7806  Fincfn 8883  cardccrd 9850

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493

This theorem is referenced by:  ackbij1lem8  10139  ackbij1lem9  10140