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Theorem cardf 9323
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 8720 . 2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
21fdmi 6014 . . . 4 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
3 cardeqv 9242 . . . 4 dom card = V
42, 3eqtr3i 2645 . . 3 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = V
54feq2i 5999 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ card:V⟶On)
61, 5mpbi 220 1 card:V⟶On
Colors of variables: wff setvar class
Syntax hints:  {cab 2607  wrex 2908  Vcvv 3189   class class class wbr 4618  dom cdm 5079  Oncon0 5687  wf 5848  cen 7903  cardccrd 8712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-ac2 9236
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-wrecs 7359  df-recs 7420  df-en 7907  df-card 8716  df-ac 8890
This theorem is referenced by:  inar1  9548
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