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Theorem cardf 10574
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 9967 . 2 card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On
21fdmi 6734 . . . 4 dom card = {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}
3 cardeqv 10493 . . . 4 dom card = V
42, 3eqtr3i 2758 . . 3 {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} = V
54feq2i 6714 . 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 2705  βˆƒwrex 3067  Vcvv 3471   class class class wbr 5148  dom cdm 5678  Oncon0 6369  βŸΆwf 6544   β‰ˆ cen 8961  cardccrd 9959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-ac2 10487
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-en 8965  df-card 9963  df-ac 10140
This theorem is referenced by:  inar1  10799
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