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Theorem cfle 9281
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
StepHypRef Expression
1 cflecard 9280 . . 3 (cf‘𝐴) ⊆ (card‘𝐴)
2 cardonle 8986 . . 3 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
31, 2syl5ss 3763 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4 cff 9275 . . . . . 6 cf:On⟶On
54fdmi 6194 . . . . 5 dom cf = On
65eleq2i 2842 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6361 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
86, 7sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
9 0ss 4117 . . 3 ∅ ⊆ 𝐴
108, 9syl6eqss 3804 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
113, 10pm2.61i 176 1 (cf‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  wss 3723  c0 4063  dom cdm 5250  Oncon0 5865  cfv 6030  cardccrd 8964  cfccf 8966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-en 8113  df-card 8968  df-cf 8970
This theorem is referenced by:  cfom  9291  cfidm  9302  alephreg  9609  winafp  9724  tskcard  9808  gruina  9845
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