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Theorem cfle 10198
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 10197 . . 3 (cfβ€˜π΄) βŠ† (cardβ€˜π΄)
2 cardonle 9901 . . 3 (𝐴 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
31, 2sstrid 3959 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† 𝐴)
4 cff 10192 . . . . . 6 cf:On⟢On
54fdmi 6684 . . . . 5 dom cf = On
65eleq2i 2826 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6881 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
86, 7sylnbir 331 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
9 0ss 4360 . . 3 βˆ… βŠ† 𝐴
108, 9eqsstrdi 4002 . 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 1542   ∈ wcel 2107   βŠ† wss 3914  βˆ…c0 4286  dom cdm 5637  Oncon0 6321  β€˜cfv 6500  cardccrd 9879  cfccf 9881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8890  df-card 9883  df-cf 9885
This theorem is referenced by:  cfom  10208  cfidm  10219  alephreg  10526  winafp  10641  tskcard  10725  gruina  10762
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