Theorem ackbij1lem7 10116

Description: Lemma for ackbij1 10128. (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 4956	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
2	1	fveq2d 6826	. 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 6835	. 2 ⊢ (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
5	2, 3, 4	fvmpt 6929	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∩ cin 3896  𝒫 cpw 4547  {csn 4573  ∪ ciun 4939   ↦ cmpt 5170   × cxp 5612  ‘cfv 6481  ωcom 7796  Fincfn 8869  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  ackbij1lem8  10117  ackbij1lem9  10118