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Theorem cfle 9028
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 9027 . . 3 (cf‘𝐴) ⊆ (card‘𝐴)
2 cardonle 8735 . . 3 (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)
31, 2syl5ss 3598 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ 𝐴)
4 cff 9022 . . . . . 6 cf:On⟶On
54fdmi 6014 . . . . 5 dom cf = On
65eleq2i 2690 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
7 ndmfv 6180 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
86, 7sylnbir 321 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
9 0ss 3949 . . 3 ∅ ⊆ 𝐴
108, 9syl6eqss 3639 . 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 1480  wcel 1987  wss 3559  c0 3896  dom cdm 5079  Oncon0 5687  cfv 5852  cardccrd 8713  cfccf 8715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-en 7908  df-card 8717  df-cf 8719
This theorem is referenced by:  cfom  9038  cfidm  9049  alephreg  9356  winafp  9471  tskcard  9555  gruina  9592
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