Theorem ackbij1lem8 9983

Description: Lemma for ackbij1 9994. (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 4571	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	fveq2d 6778	. . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3		pweq 4549	. . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
4	3	fveq2d 6778	. . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
5	2, 4	eqeq12d 2754	. 2 ⊢ (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6		ackbij1lem4 9979	. . . 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 9982	. . . 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 3436	. . . . . 6 ⊢ 𝑎 ∈ V
11		sneq 4571	. . . . . . 7 ⊢ (𝑦 = 𝑎 → {𝑦} = {𝑎})
12		pweq 4549	. . . . . . 7 ⊢ (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
13	11, 12	xpeq12d 5620	. . . . . 6 ⊢ (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
14	10, 13	iunxsn 5020	. . . . 5 ⊢ ∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
15	14	fveq2i 6777	. . . 4 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16		vpwex 5300	. . . . . 6 ⊢ 𝒫 𝑎 ∈ V
17		xpsnen2g 8852	. . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
18	10, 16, 17	mp2an 689	. . . . 5 ⊢ ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19		carden2b 9725	. . . . 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 2766	. . 3 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
22	9, 21	eqtrdi 2794	. 2 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
23	5, 22	vtoclga 3513	1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  𝒫 cpw 4533  {csn 4561  ∪ ciun 4924   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  ‘cfv 6433  ωcom 7712   ≈ cen 8730  Fincfn 8733  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-fin 8737  df-card 9697

This theorem is referenced by:  ackbij1lem14  9989  ackbij1b  9995