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Theorem cardid2 8640
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 8639 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
2 ssrab2 3650 . . . 4 {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On
3 fvex 6098 . . . . . 6 (card‘𝐴) ∈ V
41, 3syl6eqelr 2697 . . . . 5 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
5 intex 4742 . . . . 5 ({𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
64, 5sylibr 223 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅)
7 onint 6865 . . . 4 (({𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅) → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
82, 6, 7sylancr 694 . . 3 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
91, 8eqeltrd 2688 . 2 (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴})
10 breq1 4581 . . . 4 (𝑦 = (card‘𝐴) → (𝑦𝐴 ↔ (card‘𝐴) ≈ 𝐴))
1110elrab 3331 . . 3 ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
1211simprbi 479 . 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 1977  wne 2780  {crab 2900  Vcvv 3173  wss 3540  c0 3874   cint 4405   class class class wbr 4578  dom cdm 5028  Oncon0 5626  cfv 5790  cen 7816  cardccrd 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-en 7820  df-card 8626
This theorem is referenced by:  isnum3  8641  oncardid  8643  cardidm  8646  ficardom  8648  ficardid  8649  cardnn  8650  cardnueq0  8651  carden2a  8653  carden2b  8654  carddomi2  8657  sdomsdomcardi  8658  cardsdomelir  8660  cardsdomel  8661  infxpidm2  8701  dfac8b  8715  numdom  8722  alephnbtwn2  8756  alephsucdom  8763  infenaleph  8775  dfac12r  8829  cardacda  8881  pwsdompw  8887  cff1  8941  cfflb  8942  cflim2  8946  cfss  8948  cfslb  8949  domtriomlem  9125  cardid  9226  cardidg  9227  carden  9230  sdomsdomcard  9239  hargch  9352  gch2  9354  hashkf  12939
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