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Theorem cfle 9278
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 9277 . . 3 (cf‘𝐴) ⊆ (card‘𝐴)
2 cardonle 8983 . . 3 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
31, 2syl5ss 3763 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4 cff 9272 . . . . . 6 cf:On⟶On
54fdmi 6192 . . . . 5 dom cf = On
65eleq2i 2842 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6359 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
86, 7sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
9 0ss 4116 . . 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 5249  Oncon0 5866  cfv 6031  cardccrd 8961  cfccf 8963
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-en 8110  df-card 8965  df-cf 8967
This theorem is referenced by:  cfom  9288  cfidm  9299  alephreg  9606  winafp  9721  tskcard  9805  gruina  9842
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