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Theorem ackbij1lem8 9241
Description: Lemma for ackbij1 9252. (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 4331 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21fveq2d 6356 . . 3 (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3 pweq 4305 . . . 4 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
43fveq2d 6356 . . 3 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
52, 4eqeq12d 2775 . 2 (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6 ackbij1lem4 9237 . . . 4 (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
87ackbij1lem7 9240 . . . 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 3343 . . . . . 6 𝑎 ∈ V
11 sneq 4331 . . . . . . 7 (𝑦 = 𝑎 → {𝑦} = {𝑎})
12 pweq 4305 . . . . . . 7 (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
1311, 12xpeq12d 5297 . . . . . 6 (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
1410, 13iunxsn 4755 . . . . 5 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
1514fveq2i 6355 . . . 4 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16 vpwex 4998 . . . . . 6 𝒫 𝑎 ∈ V
17 xpsnen2g 8218 . . . . . 6 ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
1810, 16, 17mp2an 710 . . . . 5 ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19 carden2b 8983 . . . . 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 2782 . . 3 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
229, 21syl6eq 2810 . 2 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
235, 22vtoclga 3412 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  𝒫 cpw 4302  {csn 4321   ciun 4672   class class class wbr 4804  cmpt 4881   × cxp 5264  cfv 6049  ωcom 7230  cen 8118  Fincfn 8121  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1st 7333  df-2nd 7334  df-1o 7729  df-er 7911  df-en 8122  df-fin 8125  df-card 8955
This theorem is referenced by:  ackbij1lem14  9247  ackbij1b  9253
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