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Theorem cardcf 9666
 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 9661 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 vex 3496 . . . . . . . . 9 𝑣 ∈ V
3 eqeq1 2823 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
43anbi1d 631 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
54exbidv 1915 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
62, 5elab 3665 . . . . . . . 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 9380 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
97, 8syl6eq 2870 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
10 eqeq2 2831 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
119, 10mpbird 259 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1211adantr 483 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1312exlimiv 1924 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
146, 13sylbi 219 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
15 cardon 9365 . . . . . . 7 (card‘𝑣) ∈ On
1614, 15eqeltrrdi 2920 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1716ssriv 3969 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
18 fvex 6676 . . . . . . 7 (cf‘𝐴) ∈ V
191, 18eqeltrrdi 2920 . . . . . 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 236 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅)
22 onint 7502 . . . . 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 589 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
241, 23eqeltrd 2911 . . 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 2835 . . . 4 (𝑣 = (cf‘𝐴) → ((card‘𝑣) = 𝑣 ↔ (card‘(cf‘𝐴)) = (cf‘𝐴)))
2827, 14vtoclga 3572 . . 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 9662 . . . . . 6 cf:On⟶On
3130fdmi 6517 . . . . 5 dom cf = On
3231eleq2i 2902 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
33 ndmfv 6693 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3432, 33sylnbir 333 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
35 card0 9379 . . . 4 (card‘∅) = ∅
36 fveq2 6663 . . . 4 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (card‘∅))
37 id 22 . . . 4 ((cf‘𝐴) = ∅ → (cf‘𝐴) = ∅)
3835, 36, 373eqtr4a 2880 . . 3 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (cf‘𝐴))
3934, 38syl 17 . 2 𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
4029, 39pm2.61i 184 1 (card‘(cf‘𝐴)) = (cf‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2797   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ⊆ wss 3934  ∅c0 4289  ∩ cint 4867  dom cdm 5548  Oncon0 6184  ‘cfv 6348  cardccrd 9356  cfccf 9358 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 8281  df-en 8502  df-card 9360  df-cf 9362 This theorem is referenced by:  cfon  9669  winacard  10106
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