Theorem cfle 9076

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 9075	. . 3 ⊢ (cf‘𝐴) ⊆ (card‘𝐴)
2		cardonle 8783	. . 3 ⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
3	1, 2	syl5ss 3614	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4		cff 9070	. . . . . 6 ⊢ cf:On⟶On
5	4	fdmi 6052	. . . . 5 ⊢ dom cf = On
6	5	eleq2i 2693	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7		ndmfv 6218	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
8	6, 7	sylnbir 321	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
9		0ss 3972	. . 3 ⊢ ∅ ⊆ 𝐴
10	8, 9	syl6eqss 3655	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
11	3, 10	pm2.61i 176	1 ⊢ (cf‘𝐴) ⊆ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-card 8765  df-cf 8767

This theorem is referenced by:  cfom  9086  cfidm  9097  alephreg  9404  winafp  9519  tskcard  9603  gruina  9640