Theorem ackbij1lem8 10264

Description: Lemma for ackbij1 10275. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Hypothesis

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

Assertion

Ref	Expression
ackbij1lem8	⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))

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

Proof of Theorem ackbij1lem8

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4641	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	fveq2d 6911	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3		pweq 4619	. . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
4	3	fveq2d 6911	. . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
5	2, 4	eqeq12d 2751	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6		ackbij1lem4 10260	. . . 4 ⊢ (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7		ackbij.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
8	7	ackbij1lem7 10263	. . . 4 ⊢ ({𝑎} ∈ (𝒫 ω ∩ Fin) → (𝐹‘{𝑎}) = (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)))
9	6, 8	syl 17	. . 3 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)))
10		vex 3482	. . . . . 6 ⊢ 𝑎 ∈ V
11		sneq 4641	. . . . . . 7 ⊢ (𝑦 = 𝑎 → {𝑦} = {𝑎})
12		pweq 4619	. . . . . . 7 ⊢ (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
13	11, 12	xpeq12d 5720	. . . . . 6 ⊢ (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
14	10, 13	iunxsn 5096	. . . . 5 ⊢ ∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
15	14	fveq2i 6910	. . . 4 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16		vpwex 5383	. . . . . 6 ⊢ 𝒫 𝑎 ∈ V
17		xpsnen2g 9104	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
18	10, 16, 17	mp2an 692	. . . . 5 ⊢ ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19		carden2b 10005	. . . . 5 ⊢ (({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 → (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎))
20	18, 19	ax-mp 5	. . . 4 ⊢ (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎)
21	15, 20	eqtri 2763	. . 3 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
22	9, 21	eqtrdi 2791	. 2 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
23	5, 22	vtoclga 3577	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962  𝒫 cpw 4605  {csn 4631  ∪ ciun 4996   class class class wbr 5148   ↦ cmpt 5231   × cxp 5687  ‘cfv 6563  ωcom 7887   ≈ cen 8981  Fincfn 8984  cardccrd 9973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-en 8985  df-fin 8988  df-card 9977

This theorem is referenced by:  ackbij1lem14  10270  ackbij1b  10276