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Theorem cardcf 9662
Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cardcf (card‘(cf‘𝐴)) = (cf‘𝐴)

Proof of Theorem cardcf
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 9657 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 vex 3495 . . . . . . . . 9 𝑣 ∈ V
3 eqeq1 2822 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
43anbi1d 629 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
54exbidv 1913 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
62, 5elab 3664 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
7 fveq2 6663 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
8 cardidm 9376 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
97, 8syl6eq 2869 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
10 eqeq2 2830 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
119, 10mpbird 258 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1211adantr 481 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1312exlimiv 1922 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
146, 13sylbi 218 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
15 cardon 9361 . . . . . . 7 (card‘𝑣) ∈ On
1614, 15syl6eqelr 2919 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1716ssriv 3968 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
18 fvex 6676 . . . . . . 7 (cf‘𝐴) ∈ V
191, 18syl6eqelr 2919 . . . . . 6 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
20 intex 5231 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
2119, 20sylibr 235 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅)
22 onint 7499 . . . . 5 (({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2317, 21, 22sylancr 587 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
241, 23eqeltrd 2910 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
25 fveq2 6663 . . . . 5 (𝑣 = (cf‘𝐴) → (card‘𝑣) = (card‘(cf‘𝐴)))
26 id 22 . . . . 5 (𝑣 = (cf‘𝐴) → 𝑣 = (cf‘𝐴))
2725, 26eqeq12d 2834 . . . 4 (𝑣 = (cf‘𝐴) → ((card‘𝑣) = 𝑣 ↔ (card‘(cf‘𝐴)) = (cf‘𝐴)))
2827, 14vtoclga 3571 . . 3 ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘(cf‘𝐴)) = (cf‘𝐴))
2924, 28syl 17 . 2 (𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
30 cff 9658 . . . . . 6 cf:On⟶On
3130fdmi 6517 . . . . 5 dom cf = On
3231eleq2i 2901 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
33 ndmfv 6693 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3432, 33sylnbir 332 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
35 card0 9375 . . . 4 (card‘∅) = ∅
36 fveq2 6663 . . . 4 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (card‘∅))
37 id 22 . . . 4 ((cf‘𝐴) = ∅ → (cf‘𝐴) = ∅)
3835, 36, 373eqtr4a 2879 . . 3 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (cf‘𝐴))
3934, 38syl 17 . 2 𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
4029, 39pm2.61i 183 1 (card‘(cf‘𝐴)) = (cf‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288   cint 4867  dom cdm 5548  Oncon0 6184  cfv 6348  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-er 8278  df-en 8498  df-card 9356  df-cf 9358
This theorem is referenced by:  cfon  9665  winacard  10102
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