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Theorem cardcf 9362
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 9357 . . . 4 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 vex 3388 . . . . . . . . 9 𝑣 ∈ V
3 eqeq1 2803 . . . . . . . . . . 11 (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
43anbi1d 624 . . . . . . . . . 10 (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
54exbidv 2017 . . . . . . . . 9 (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
62, 5elab 3542 . . . . . . . 8 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
7 fveq2 6411 . . . . . . . . . . . 12 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
8 cardidm 9071 . . . . . . . . . . . 12 (card‘(card‘𝑦)) = (card‘𝑦)
97, 8syl6eq 2849 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
10 eqeq2 2810 . . . . . . . . . . 11 (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
119, 10mpbird 249 . . . . . . . . . 10 (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
1211adantr 473 . . . . . . . . 9 ((𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
1312exlimiv 2026 . . . . . . . 8 (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (card‘𝑣) = 𝑣)
146, 13sylbi 209 . . . . . . 7 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → (card‘𝑣) = 𝑣)
15 cardon 9056 . . . . . . 7 (card‘𝑣) ∈ On
1614, 15syl6eqelr 2887 . . . . . 6 (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → 𝑣 ∈ On)
1716ssriv 3802 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ On
18 fvex 6424 . . . . . . 7 (cf‘𝐴) ∈ V
191, 18syl6eqelr 2887 . . . . . 6 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
20 intex 5012 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅ ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
2119, 20sylibr 226 . . . . 5 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ≠ ∅)
22 onint 7229 . . . . 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 582 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
241, 23eqeltrd 2878 . . 3 (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
25 fveq2 6411 . . . . 5 (𝑣 = (cf‘𝐴) → (card‘𝑣) = (card‘(cf‘𝐴)))
26 id 22 . . . . 5 (𝑣 = (cf‘𝐴) → 𝑣 = (cf‘𝐴))
2725, 26eqeq12d 2814 . . . 4 (𝑣 = (cf‘𝐴) → ((card‘𝑣) = 𝑣 ↔ (card‘(cf‘𝐴)) = (cf‘𝐴)))
2827, 14vtoclga 3460 . . 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 9358 . . . . . 6 cf:On⟶On
3130fdmi 6266 . . . . 5 dom cf = On
3231eleq2i 2870 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
33 ndmfv 6441 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3432, 33sylnbir 323 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
35 card0 9070 . . . 4 (card‘∅) = ∅
36 fveq2 6411 . . . 4 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (card‘∅))
37 id 22 . . . 4 ((cf‘𝐴) = ∅ → (cf‘𝐴) = ∅)
3835, 36, 373eqtr4a 2859 . . 3 ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (cf‘𝐴))
3934, 38syl 17 . 2 𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
4029, 39pm2.61i 177 1 (card‘(cf‘𝐴)) = (cf‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wne 2971  wral 3089  wrex 3090  Vcvv 3385  wss 3769  c0 4115   cint 4667  dom cdm 5312  Oncon0 5941  cfv 6101  cardccrd 9047  cfccf 9049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-er 7982  df-en 8196  df-card 9051  df-cf 9053
This theorem is referenced by:  cfon  9365  winacard  9802
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