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Theorem cardid2 9944
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardid2 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)

Proof of Theorem cardid2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cardval3 9943 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
2 ssrab2 4076 . . . 4 {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On
3 fvex 6901 . . . . . 6 (card‘𝐴) ∈ V
41, 3eqeltrrdi 2842 . . . . 5 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
5 intex 5336 . . . . 5 ({𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
64, 5sylibr 233 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅)
7 onint 7774 . . . 4 (({𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅) → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
82, 6, 7sylancr 587 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
91, 8eqeltrd 2833 . 2 (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
10 breq1 5150 . . . 4 (𝑦 = (card‘𝐴) → (𝑦𝐴 ↔ (card‘𝐴) ≈ 𝐴))
1110elrab 3682 . . 3 ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
1211simprbi 497 . 2 ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴} → (card‘𝐴) ≈ 𝐴)
139, 12syl 17 1 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2940  {crab 3432  Vcvv 3474  wss 3947  c0 4321   cint 4949   class class class wbr 5147  dom cdm 5675  Oncon0 6361  cfv 6540  cen 8932  cardccrd 9926
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-en 8936  df-card 9930
This theorem is referenced by:  isnum3  9945  oncardid  9947  cardidm  9950  ficardom  9952  ficardid  9953  cardnn  9954  cardnueq0  9955  carden2a  9957  carden2b  9958  carddomi2  9961  sdomsdomcardi  9962  cardsdomelir  9964  cardsdomel  9965  infxpidm2  10008  dfac8b  10022  numdom  10029  alephnbtwn2  10063  alephsucdom  10070  infenaleph  10082  dfac12r  10137  cardadju  10185  pwsdompw  10195  cff1  10249  cfflb  10250  cflim2  10254  cfss  10256  cfslb  10257  domtriomlem  10433  cardid  10538  cardidg  10539  carden  10542  sdomsdomcard  10551  hargch  10664  gch2  10666  hashkf  14288
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