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Theorem cardf 9963
 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 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 9358 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
21fdmi 6498 . . . 4 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
3 cardeqv 9882 . . . 4 dom card = V
42, 3eqtr3i 2823 . . 3 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = V
54feq2i 6479 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ card:V⟶On)
61, 5mpbi 233 1 card:V⟶On
 Colors of variables: wff setvar class Syntax hints:  {cab 2776  ∃wrex 3107  Vcvv 3441   class class class wbr 5030  dom cdm 5519  Oncon0 6159  ⟶wf 6320   ≈ cen 8491  cardccrd 9350 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-ac2 9876 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-wrecs 7932  df-recs 7993  df-en 8495  df-card 9354  df-ac 9529 This theorem is referenced by:  inar1  10188
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