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

Theorem ackbij1lem8 9034
Description: Lemma for ackbij1 9045. (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 4178 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21fveq2d 6182 . . 3 (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3 pweq 4152 . . . 4 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
43fveq2d 6182 . . 3 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
52, 4eqeq12d 2635 . 2 (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6 ackbij1lem4 9030 . . . 4 (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
87ackbij1lem7 9033 . . . 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 3198 . . . . . 6 𝑎 ∈ V
11 sneq 4178 . . . . . . 7 (𝑦 = 𝑎 → {𝑦} = {𝑎})
12 pweq 4152 . . . . . . 7 (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
1311, 12xpeq12d 5130 . . . . . 6 (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
1410, 13iunxsn 4594 . . . . 5 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
1514fveq2i 6181 . . . 4 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16 vpwex 4840 . . . . . 6 𝒫 𝑎 ∈ V
17 xpsnen2g 8038 . . . . . 6 ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
1810, 16, 17mp2an 707 . . . . 5 ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19 carden2b 8778 . . . . 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 2642 . . 3 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
229, 21syl6eq 2670 . 2 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
235, 22vtoclga 3267 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  𝒫 cpw 4149  {csn 4168   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  cfv 5876  ωcom 7050  cen 7937  Fincfn 7940  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1st 7153  df-2nd 7154  df-1o 7545  df-er 7727  df-en 7941  df-fin 7944  df-card 8750
This theorem is referenced by:  ackbij1lem14  9040  ackbij1b  9046
  Copyright terms: Public domain W3C validator