Theorem cardf2 8769

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, 20-Sep-2014.)

Assertion

Ref	Expression
cardf2	|- card : { x | E. y e. On y ~~ x } --> On

Distinct variable group:   x, y

Proof of Theorem cardf2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-card 8765	. . . 4 |- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
2	1	funmpt2 5927	. . 3 |- Fun card
3		rabab 3223	. . . 4 |- { x e. _V | |^| { y e. On | y ~~ x } e. _V } = { x | |^| { y e. On | y ~~ x } e. _V }
4	1	dmmpt 5630	. . . 4 |- dom card = { x e. _V | |^| { y e. On | y ~~ x } e. _V }
5		intexrab 4823	. . . . 5 |- ( E. y e. On y ~~ x <-> |^| { y e. On | y ~~ x } e. _V )
6	5	abbii 2739	. . . 4 |- { x | E. y e. On y ~~ x } = { x | |^| { y e. On | y ~~ x } e. _V }
7	3, 4, 6	3eqtr4i 2654	. . 3 |- dom card = { x | E. y e. On y ~~ x }
8		df-fn 5891	. . 3 |- ( card Fn { x | E. y e. On y ~~ x } <-> ( Fun card /\ dom card = { x | E. y e. On y ~~ x } ) )
9	2, 7, 8	mpbir2an 955	. 2 |- card Fn { x | E. y e. On y ~~ x }
10		simpr 477	. . . . . . . . 9 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> w = |^| { y e. On | y ~~ z } )
11		vex 3203	. . . . . . . . 9 |- w e. _V
12	10, 11	syl6eqelr 2710	. . . . . . . 8 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> |^| { y e. On | y ~~ z } e. _V )
13		intex 4820	. . . . . . . 8 |- ( { y e. On | y ~~ z } =/= (/) <-> |^| { y e. On | y ~~ z } e. _V )
14	12, 13	sylibr 224	. . . . . . 7 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> { y e. On | y ~~ z } =/= (/) )
15		rabn0 3958	. . . . . . 7 |- ( { y e. On | y ~~ z } =/= (/) <-> E. y e. On y ~~ z )
16	14, 15	sylib 208	. . . . . 6 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> E. y e. On y ~~ z )
17		vex 3203	. . . . . . 7 |- z e. _V
18		breq2 4657	. . . . . . . 8 |- ( x = z -> ( y ~~ x <-> y ~~ z ) )
19	18	rexbidv 3052	. . . . . . 7 |- ( x = z -> ( E. y e. On y ~~ x <-> E. y e. On y ~~ z ) )
20	17, 19	elab 3350	. . . . . 6 |- ( z e. { x | E. y e. On y ~~ x } <-> E. y e. On y ~~ z )
21	16, 20	sylibr 224	. . . . 5 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> z e. { x | E. y e. On y ~~ x } )
22		ssrab2 3687	. . . . . . 7 |- { y e. On | y ~~ z } C_ On
23		oninton 7000	. . . . . . 7 |- ( ( { y e. On | y ~~ z } C_ On /\ { y e. On | y ~~ z } =/= (/) ) -> |^| { y e. On | y ~~ z } e. On )
24	22, 14, 23	sylancr 695	. . . . . 6 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> |^| { y e. On | y ~~ z } e. On )
25	10, 24	eqeltrd 2701	. . . . 5 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> w e. On )
26	21, 25	jca 554	. . . 4 |- ( ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) -> ( z e. { x | E. y e. On y ~~ x } /\ w e. On ) )
27	26	ssopab2i 5003	. . 3 |- { <. z , w >. | ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) } C_ { <. z , w >. | ( z e. { x | E. y e. On y ~~ x } /\ w e. On ) }
28		df-card 8765	. . . 4 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
29		df-mpt 4730	. . . 4 |- ( z e. _V |-> |^| { y e. On | y ~~ z } ) = { <. z , w >. | ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) }
30	28, 29	eqtri 2644	. . 3 |- card = { <. z , w >. | ( z e. _V /\ w = |^| { y e. On | y ~~ z } ) }
31		df-xp 5120	. . 3 |- ( { x | E. y e. On y ~~ x } X. On ) = { <. z , w >. | ( z e. { x | E. y e. On y ~~ x } /\ w e. On ) }
32	27, 30, 31	3sstr4i 3644	. 2 |- card C_ ( { x | E. y e. On y ~~ x } X. On )
33		dff2 6371	. 2 |- ( card : { x | E. y e. On y ~~ x } --> On <-> ( card Fn { x | E. y e. On y ~~ x } /\ card C_ ( { x | E. y e. On y ~~ x } X. On ) ) )
34	9, 32, 33	mpbir2an 955	1 |- card : { x | E. y e. On y ~~ x } --> On

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ~~ cen 7952   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-fun 5890  df-fn 5891  df-f 5892  df-card 8765

This theorem is referenced by:  cardon  8770  isnum2  8771  cardf  9372  smobeth  9408  hashkf  13119  hashgval  13120