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Theorem cardf 10542
Description: The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
cardf card:V⟢On

Proof of Theorem cardf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 9935 . 2 card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On
21fdmi 6720 . . . 4 dom card = {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}
3 cardeqv 10461 . . . 4 dom card = V
42, 3eqtr3i 2754 . . 3 {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} = V
54feq2i 6700 . 2 (card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On ↔ card:V⟢On)
61, 5mpbi 229 1 card:V⟢On
Colors of variables: wff setvar class
Syntax hints:  {cab 2701  βˆƒwrex 3062  Vcvv 3466   class class class wbr 5139  dom cdm 5667  Oncon0 6355  βŸΆwf 6530   β‰ˆ cen 8933  cardccrd 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-ac2 10455
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-en 8937  df-card 9931  df-ac 10108
This theorem is referenced by:  inar1  10767
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