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Theorem cfval2 9270
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 9257 . 2 (𝐴 ∈ On → (cf‘𝐴) = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
2 fvex 6358 . . . 4 (card‘𝑥) ∈ V
32dfiin2 4703 . . 3 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)}
4 df-rex 3052 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)))
5 rabid 3250 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
6 selpw 4305 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
76anbi1i 733 . . . . . . . . 9 ((𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
85, 7bitri 264 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
98anbi2ci 734 . . . . . . 7 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
109exbii 1919 . . . . . 6 (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
114, 10bitri 264 . . . . 5 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1211abbii 2873 . . . 4 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
1312inteqi 4627 . . 3 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
143, 13eqtr2i 2779 . 2 {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))} = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥)
151, 14syl6eq 2806 1 (𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wex 1849  wcel 2135  {cab 2742  wral 3046  wrex 3047  {crab 3050  wss 3711  𝒫 cpw 4298   cint 4623   ciin 4669  Oncon0 5880  cfv 6045  cardccrd 8947  cfccf 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-int 4624  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-iota 6008  df-fun 6047  df-fv 6053  df-cf 8953
This theorem is referenced by: (None)
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