Theorem ackbij1lem7 10316

Description: Lemma for ackbij1 10328. (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 4971	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
2	1	fveq2d 6900	. 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 6909	. 2 ⊢ (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
5	2, 3, 4	fvmpt 7004	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ ciun 4954   ↦ cmpt 5189   × cxp 5653  ‘cfv 6547  ωcom 7887  Fincfn 8987  cardccrd 10029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6501  df-fun 6549  df-fv 6555

This theorem is used by:  ackbij1lem8  10317  ackbij1lem9  10318