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Theorem cardid2 8662
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 8661 . . 3 (𝐴 ∈ dom card → (card‘𝐴) = {𝑦 ∈ On ∣ 𝑦𝐴})
2 ssrab2 3650 . . . 4 {𝑦 ∈ On ∣ 𝑦𝐴} ⊆ On
3 fvex 6113 . . . . . 6 (card‘𝐴) ∈ V
41, 3syl6eqelr 2697 . . . . 5 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
5 intex 4747 . . . . 5 ({𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝐴} ∈ V)
64, 5sylibr 223 . . . 4 (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦𝐴} ≠ ∅)
7 onint 6887 . . . 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 4586 . . . 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 4410   class class class wbr 4583  dom cdm 5038  Oncon0 5640  cfv 5804  cen 7838  cardccrd 8644
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 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
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 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-en 7842  df-card 8648
This theorem is referenced by:  isnum3  8663  oncardid  8665  cardidm  8668  ficardom  8670  ficardid  8671  cardnn  8672  cardnueq0  8673  carden2a  8675  carden2b  8676  carddomi2  8679  sdomsdomcardi  8680  cardsdomelir  8682  cardsdomel  8683  infxpidm2  8723  dfac8b  8737  numdom  8744  alephnbtwn2  8778  alephsucdom  8785  infenaleph  8797  dfac12r  8851  cardacda  8903  pwsdompw  8909  cff1  8963  cfflb  8964  cflim2  8968  cfss  8970  cfslb  8971  domtriomlem  9147  cardid  9248  cardidg  9249  carden  9252  sdomsdomcard  9261  hargch  9374  gch2  9376  hashkf  12981
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