Theorem cflecard 9075

Description: Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cflecard	|- ( cf ` A ) C_ ( card ` A )

Proof of Theorem cflecard

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

Step	Hyp	Ref	Expression
1		cfval 9069	. . 3 |- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
2		df-sn 4178	. . . . . 6 |- { ( card ` A ) } = { x | x = ( card ` A ) }
3		ssid 3624	. . . . . . . . 9 |- A C_ A
4		ssid 3624	. . . . . . . . . . 11 |- z C_ z
5		sseq2 3627	. . . . . . . . . . . 12 |- ( w = z -> ( z C_ w <-> z C_ z ) )
6	5	rspcev 3309	. . . . . . . . . . 11 |- ( ( z e. A /\ z C_ z ) -> E. w e. A z C_ w )
7	4, 6	mpan2 707	. . . . . . . . . 10 |- ( z e. A -> E. w e. A z C_ w )
8	7	rgen 2922	. . . . . . . . 9 |- A. z e. A E. w e. A z C_ w
9	3, 8	pm3.2i 471	. . . . . . . 8 |- ( A C_ A /\ A. z e. A E. w e. A z C_ w )
10		fveq2 6191	. . . . . . . . . . 11 |- ( y = A -> ( card ` y ) = ( card ` A ) )
11	10	eqeq2d 2632	. . . . . . . . . 10 |- ( y = A -> ( x = ( card ` y ) <-> x = ( card ` A ) ) )
12		sseq1 3626	. . . . . . . . . . 11 |- ( y = A -> ( y C_ A <-> A C_ A ) )
13		rexeq 3139	. . . . . . . . . . . 12 |- ( y = A -> ( E. w e. y z C_ w <-> E. w e. A z C_ w ) )
14	13	ralbidv 2986	. . . . . . . . . . 11 |- ( y = A -> ( A. z e. A E. w e. y z C_ w <-> A. z e. A E. w e. A z C_ w ) )
15	12, 14	anbi12d 747	. . . . . . . . . 10 |- ( y = A -> ( ( y C_ A /\ A. z e. A E. w e. y z C_ w ) <-> ( A C_ A /\ A. z e. A E. w e. A z C_ w ) ) )
16	11, 15	anbi12d 747	. . . . . . . . 9 |- ( y = A -> ( ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) <-> ( x = ( card ` A ) /\ ( A C_ A /\ A. z e. A E. w e. A z C_ w ) ) ) )
17	16	spcegv 3294	. . . . . . . 8 |- ( A e. On -> ( ( x = ( card ` A ) /\ ( A C_ A /\ A. z e. A E. w e. A z C_ w ) ) -> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
18	9, 17	mpan2i 713	. . . . . . 7 |- ( A e. On -> ( x = ( card ` A ) -> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
19	18	ss2abdv 3675	. . . . . 6 |- ( A e. On -> { x | x = ( card ` A ) } C_ { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
20	2, 19	syl5eqss 3649	. . . . 5 |- ( A e. On -> { ( card ` A ) } C_ { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
21		intss 4498	. . . . 5 |- ( { ( card ` A ) } C_ { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } C_ |^| { ( card ` A ) } )
22	20, 21	syl 17	. . . 4 |- ( A e. On -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } C_ |^| { ( card ` A ) } )
23		fvex 6201	. . . . 5 |- ( card ` A ) e. _V
24	23	intsn 4513	. . . 4 |- |^| { ( card ` A ) } = ( card ` A )
25	22, 24	syl6sseq 3651	. . 3 |- ( A e. On -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } C_ ( card ` A ) )
26	1, 25	eqsstrd 3639	. 2 |- ( A e. On -> ( cf ` A ) C_ ( card ` A ) )
27		cff 9070	. . . . . 6 |- cf : On --> On
28	27	fdmi 6052	. . . . 5 |- dom cf = On
29	28	eleq2i 2693	. . . 4 |- ( A e. dom cf <-> A e. On )
30		ndmfv 6218	. . . 4 |- ( -. A e. dom cf -> ( cf ` A ) = (/) )
31	29, 30	sylnbir 321	. . 3 |- ( -. A e. On -> ( cf ` A ) = (/) )
32		0ss 3972	. . 3 |- (/) C_ ( card ` A )
33	31, 32	syl6eqss 3655	. 2 |- ( -. A e. On -> ( cf ` A ) C_ ( card ` A ) )
34	26, 33	pm2.61i 176	1 |- ( cf ` A ) C_ ( card ` A )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   {csn 4177   |^|cint 4475   dom cdm 5114   Oncon0 5723   ` cfv 5888   cardccrd 8761   cfccf 8763

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-pow 4843  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-pw 4160  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-card 8765  df-cf 8767

This theorem is referenced by:  cfle  9076