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Theorem ackbij1lem8 9641
 Description: Lemma for ackbij1 9652. (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 4535 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21fveq2d 6650 . . 3 (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3 pweq 4513 . . . 4 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
43fveq2d 6650 . . 3 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
52, 4eqeq12d 2814 . 2 (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6 ackbij1lem4 9637 . . . 4 (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
87ackbij1lem7 9640 . . . 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 3444 . . . . . 6 𝑎 ∈ V
11 sneq 4535 . . . . . . 7 (𝑦 = 𝑎 → {𝑦} = {𝑎})
12 pweq 4513 . . . . . . 7 (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
1311, 12xpeq12d 5551 . . . . . 6 (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
1410, 13iunxsn 4977 . . . . 5 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
1514fveq2i 6649 . . . 4 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16 vpwex 5244 . . . . . 6 𝒫 𝑎 ∈ V
17 xpsnen2g 8596 . . . . . 6 ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
1810, 16, 17mp2an 691 . . . . 5 ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19 carden2b 9383 . . . . 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 2821 . . 3 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
229, 21eqtrdi 2849 . 2 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
235, 22vtoclga 3522 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  𝒫 cpw 4497  {csn 4525  ∪ ciun 4882   class class class wbr 5031   ↦ cmpt 5111   × cxp 5518  ‘cfv 6325  ωcom 7563   ≈ cen 8492  Fincfn 8495  cardccrd 9351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-om 7564  df-1st 7674  df-2nd 7675  df-1o 8088  df-er 8275  df-en 8496  df-fin 8499  df-card 9355 This theorem is referenced by:  ackbij1lem14  9647  ackbij1b  9653
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