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Theorem cfle 10246
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 (cfβ€˜π΄) βŠ† 𝐴

Proof of Theorem cfle
StepHypRef Expression
1 cflecard 10245 . . 3 (cfβ€˜π΄) βŠ† (cardβ€˜π΄)
2 cardonle 9949 . . 3 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
31, 2sstrid 3986 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† 𝐴)
4 cff 10240 . . . . . 6 cf:On⟢On
54fdmi 6720 . . . . 5 dom cf = On
65eleq2i 2817 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6917 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
86, 7sylnbir 331 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
9 0ss 4389 . . 3 βˆ… βŠ† 𝐴
108, 9eqsstrdi 4029 . 2 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† 𝐴)
113, 10pm2.61i 182 1 (cfβ€˜π΄) βŠ† 𝐴
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941  βˆ…c0 4315  dom cdm 5667  Oncon0 6355  β€˜cfv 6534  cardccrd 9927  cfccf 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-en 8937  df-card 9931  df-cf 9933
This theorem is referenced by:  cfom  10256  cfidm  10267  alephreg  10574  winafp  10689  tskcard  10773  gruina  10810
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