MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem8 Structured version   Visualization version   GIF version

Theorem ackbij1lem8 10266
Description: Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 19-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({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.
StepHypRef Expression
1 sneq 4636 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21fveq2d 6910 . . 3 (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3 pweq 4614 . . . 4 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
43fveq2d 6910 . . 3 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
52, 4eqeq12d 2753 . 2 (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6 ackbij1lem4 10262 . . . 4 (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
87ackbij1lem7 10265 . . . 4 ({𝑎} ∈ (𝒫 ω ∩ Fin) → (𝐹‘{𝑎}) = (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)))
96, 8syl 17 . . 3 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)))
10 vex 3484 . . . . . 6 𝑎 ∈ V
11 sneq 4636 . . . . . . 7 (𝑦 = 𝑎 → {𝑦} = {𝑎})
12 pweq 4614 . . . . . . 7 (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
1311, 12xpeq12d 5716 . . . . . 6 (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
1410, 13iunxsn 5091 . . . . 5 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
1514fveq2i 6909 . . . 4 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16 vpwex 5377 . . . . . 6 𝒫 𝑎 ∈ V
17 xpsnen2g 9105 . . . . . 6 ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
1810, 16, 17mp2an 692 . . . . 5 ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19 carden2b 10007 . . . . 5 (({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 → (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎))
2018, 19ax-mp 5 . . . 4 (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎)
2115, 20eqtri 2765 . . 3 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
229, 21eqtrdi 2793 . 2 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
235, 22vtoclga 3577 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  𝒫 cpw 4600  {csn 4626   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  cfv 6561  ωcom 7887  cen 8982  Fincfn 8985  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-en 8986  df-fin 8989  df-card 9979
This theorem is referenced by:  ackbij1lem14  10272  ackbij1b  10278
  Copyright terms: Public domain W3C validator