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Theorem cfle 8959
 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 8958 . . 3 (cf‘𝐴) ⊆ (card‘𝐴)
2 cardonle 8666 . . 3 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
31, 2syl5ss 3579 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4 cff 8953 . . . . . 6 cf:On⟶On
54fdmi 5965 . . . . 5 dom cf = On
65eleq2i 2680 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6128 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
86, 7sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
9 0ss 3924 . . 3 ∅ ⊆ 𝐴
108, 9syl6eqss 3618 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
113, 10pm2.61i 175 1 (cf‘𝐴) ⊆ 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  dom cdm 5038  Oncon0 5640  ‘cfv 5804  cardccrd 8644  cfccf 8646 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-en 7842  df-card 8648  df-cf 8650 This theorem is referenced by:  cfom  8969  cfidm  8980  alephreg  9283  winafp  9398  tskcard  9482  gruina  9519
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