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Theorem cardid2 9370
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 9369 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
2 ssrab2 4053 . . . 4 {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On
3 fvex 6676 . . . . . 6 (card‘𝐴) ∈ V
41, 3syl6eqelr 2919 . . . . 5 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
5 intex 5231 . . . . 5 ({𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
64, 5sylibr 235 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅)
7 onint 7499 . . . 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 2910 . 2 (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
10 breq1 5060 . . . 4 (𝑦 = (card‘𝐴) → (𝑦𝐴 ↔ (card‘𝐴) ≈ 𝐴))
1110elrab 3677 . . 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 2105  wne 3013  {crab 3139  Vcvv 3492  wss 3933  c0 4288   cint 4867   class class class wbr 5057  dom cdm 5548  Oncon0 6184  cfv 6348  cen 8494  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-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-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  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-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-en 8498  df-card 9356
This theorem is referenced by:  isnum3  9371  oncardid  9373  cardidm  9376  ficardom  9378  ficardid  9379  cardnn  9380  cardnueq0  9381  carden2a  9383  carden2b  9384  carddomi2  9387  sdomsdomcardi  9388  cardsdomelir  9390  cardsdomel  9391  infxpidm2  9431  dfac8b  9445  numdom  9452  alephnbtwn2  9486  alephsucdom  9493  infenaleph  9505  dfac12r  9560  cardadju  9608  pwsdompw  9614  cff1  9668  cfflb  9669  cflim2  9673  cfss  9675  cfslb  9676  domtriomlem  9852  cardid  9957  cardidg  9958  carden  9961  sdomsdomcard  9970  hargch  10083  gch2  10085  hashkf  13680
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