Theorem cfle 9941

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 9940	. . 3 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)
2		cardonle 9646	. . 3 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3	1, 2	sstrid 3928	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4		cff 9935	. . . . . 6 ⊢ cf:On⟶On
5	4	fdmi 6596	. . . . 5 ⊢ dom cf = On
6	5	eleq2i 2830	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7		ndmfv 6786	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
8	6, 7	sylnbir 330	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
9		0ss 4327	. . 3 ⊢ ∅ ⊆ 𝐴
10	8, 9	eqsstrdi 3971	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
11	3, 10	pm2.61i 182	1 ⊢ (cf‘𝐴) ⊆ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ∅c0 4253  dom cdm 5580  Oncon0 6251  ‘cfv 6418  cardccrd 9624  cfccf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-en 8692  df-card 9628  df-cf 9630

This theorem is referenced by:  cfom  9951  cfidm  9962  alephreg  10269  winafp  10384  tskcard  10468  gruina  10505