Theorem ackbij1lem7 10303

Description: Lemma for ackbij1 10315. (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 4968	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
2	1	fveq2d 6889	. 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 6898	. 2 ⊢ (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V
5	2, 3, 4	fvmpt 6993	1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∩ cin 3898  𝒫 cpw 4557  {csn 4584  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5649  ‘cfv 6538  ωcom 7877  Fincfn 8973  cardccrd 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  ackbij1lem8  10304  ackbij1lem9  10305