Theorem cardf 10575

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.

Step	Hyp	Ref	Expression
1		cardf2 9968	. 2 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On
2	1	fdmi 6734	. . . 4 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
3		cardeqv 10494	. . . 4 ⊢ dom card = V
4	2, 3	eqtr3i 2755	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = V
5	4	feq2i 6715	. 2 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On ↔ card:V⟶On)
6	1, 5	mpbi 229	1 ⊢ card:V⟶On

Syntax hints:  {cab 2702  ∃wrex 3059  Vcvv 3461   class class class wbr 5149  dom cdm 5678  Oncon0 6371  ⟶wf 6545   ≈ cen 8961  cardccrd 9960

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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-ac2 10488

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6307  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-en 8965  df-card 9964  df-ac 10141

This theorem is referenced by:  inar1  10800