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Theorem cfval2 9720
 Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cfval2 (𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))
Distinct variable group:   𝑤,𝐴,𝑥,𝑧

Proof of Theorem cfval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cfval 9707 . 2 (𝐴 ∈ On → (cf‘𝐴) = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
2 fvex 6671 . . . 4 (card‘𝑥) ∈ V
32dfiin2 4923 . . 3 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)}
4 df-rex 3076 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)))
5 rabid 3296 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
6 velpw 4499 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
76anbi1i 626 . . . . . . . . 9 ((𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
85, 7bitri 278 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
98anbi2ci 627 . . . . . . 7 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
109exbii 1849 . . . . . 6 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
114, 10bitri 278 . . . . 5 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1211abbii 2823 . . . 4 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
1312inteqi 4842 . . 3 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
143, 13eqtr2i 2782 . 2 {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))} = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥)
151, 14eqtrdi 2809 1 (𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  {crab 3074   ⊆ wss 3858  𝒫 cpw 4494  ∩ cint 4838  ∩ ciin 4884  Oncon0 6169  ‘cfv 6335  cardccrd 9397  cfccf 9399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-cf 9403 This theorem is referenced by: (None)
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