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Theorem ackbij1lem8 9637
Description: Lemma for ackbij1 9648. (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 4567 . . . 4 (𝑎 = 𝐴 → {𝑎} = {𝐴})
21fveq2d 6667 . . 3 (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴}))
3 pweq 4538 . . . 4 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
43fveq2d 6667 . . 3 (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴))
52, 4eqeq12d 2834 . 2 (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴)))
6 ackbij1lem4 9633 . . . 4 (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin))
7 ackbij.f . . . . 5 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
87ackbij1lem7 9636 . . . 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 3495 . . . . . 6 𝑎 ∈ V
11 sneq 4567 . . . . . . 7 (𝑦 = 𝑎 → {𝑦} = {𝑎})
12 pweq 4538 . . . . . . 7 (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎)
1311, 12xpeq12d 5579 . . . . . 6 (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎))
1410, 13iunxsn 5004 . . . . 5 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)
1514fveq2i 6666 . . . 4 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎))
16 vpwex 5269 . . . . . 6 𝒫 𝑎 ∈ V
17 xpsnen2g 8598 . . . . . 6 ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎)
1810, 16, 17mp2an 688 . . . . 5 ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎
19 carden2b 9384 . . . . 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 2841 . . 3 (card‘ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎)
229, 21syl6eq 2869 . 2 (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎))
235, 22vtoclga 3571 1 (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cin 3932  𝒫 cpw 4535  {csn 4557   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  cfv 6348  ωcom 7569  cen 8494  Fincfn 8497  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1st 7678  df-2nd 7679  df-1o 8091  df-er 8278  df-en 8498  df-fin 8501  df-card 9356
This theorem is referenced by:  ackbij1lem14  9643  ackbij1b  9649
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