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Theorem cfval2 9026
 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 9013 . 2 (𝐴 ∈ On → (cf‘𝐴) = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
2 fvex 6158 . . . 4 (card‘𝑥) ∈ V
32dfiin2 4521 . . 3 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)}
4 df-rex 2913 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)))
5 rabid 3106 . . . . . . . . 9 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
6 selpw 4137 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
76anbi1i 730 . . . . . . . . 9 ((𝑥 ∈ 𝒫 𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
85, 7bitri 264 . . . . . . . 8 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
98anbi2ci 731 . . . . . . 7 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
109exbii 1771 . . . . . 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 2736 . . . 4 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
1312inteqi 4444 . . 3 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))}
143, 13eqtr2i 2644 . 2 {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))} = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥)
151, 14syl6eq 2671 1 (𝐴 ∈ On → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧𝐴𝑤𝑥 𝑧𝑤} (card‘𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3555  𝒫 cpw 4130  ∩ cint 4440  ∩ ciin 4486  Oncon0 5682  ‘cfv 5847  cardccrd 8705  cfccf 8707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-cf 8711 This theorem is referenced by: (None)
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