Theorem cfle 9665

Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cfle	⊢ (cf‘𝐴) ⊆ 𝐴

Proof of Theorem cfle

Step	Hyp	Ref	Expression
1		cflecard 9664	. . 3 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)
2		cardonle 9370	. . 3 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3	1, 2	sstrid 3926	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4		cff 9659	. . . . . 6 ⊢ cf:On⟶On
5	4	fdmi 6498	. . . . 5 ⊢ dom cf = On
6	5	eleq2i 2881	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7		ndmfv 6675	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
8	6, 7	sylnbir 334	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
9		0ss 4304	. . 3 ⊢ ∅ ⊆ 𝐴
10	8, 9	eqsstrdi 3969	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
11	3, 10	pm2.61i 185	1 ⊢ (cf‘𝐴) ⊆ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  dom cdm 5519  Oncon0 6159  ‘cfv 6324  cardccrd 9348  cfccf 9350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-en 8493  df-card 9352  df-cf 9354

This theorem is referenced by:  cfom  9675  cfidm  9686  alephreg  9993  winafp  10108  tskcard  10192  gruina  10229