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Theorem cfle 9665
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 9664 . . 3 (cf‘𝐴) ⊆ (card‘𝐴)
2 cardonle 9374 . . 3 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
31, 2sstrid 3953 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4 cff 9659 . . . . . 6 cf:On⟶On
54fdmi 6505 . . . . 5 dom cf = On
65eleq2i 2905 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6682 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
86, 7sylnbir 334 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
9 0ss 4322 . . 3 ∅ ⊆ 𝐴
108, 9eqsstrdi 3996 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
113, 10pm2.61i 185 1 (cf‘𝐴) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2114  wss 3908  c0 4265  dom cdm 5532  Oncon0 6169  cfv 6334  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-en 8497  df-card 9356  df-cf 9358
This theorem is referenced by:  cfom  9675  cfidm  9686  alephreg  9993  winafp  10108  tskcard  10192  gruina  10229
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