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Theorem cflim3 10300
Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
Hypothesis
Ref Expression
cflim3.1 𝐴 ∈ V
Assertion
Ref Expression
cflim3 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem cflim3
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limord 6446 . . . 4 (Lim 𝐴 → Ord 𝐴)
2 cflim3.1 . . . . 5 𝐴 ∈ V
32elon 6395 . . . 4 (𝐴 ∈ On ↔ Ord 𝐴)
41, 3sylibr 234 . . 3 (Lim 𝐴𝐴 ∈ On)
5 cfval 10285 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
64, 5syl 17 . 2 (Lim 𝐴 → (cf‘𝐴) = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
7 fvex 6920 . . . 4 (card‘𝑥) ∈ V
87dfiin2 5039 . . 3 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9 df-rex 3069 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10 ancom 460 . . . . . . . 8 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}))
11 rabid 3455 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
12 velpw 4610 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1312anbi1i 624 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 𝑥 = 𝐴))
14 coflim 10299 . . . . . . . . . . . 12 ((Lim 𝐴𝑥𝐴) → ( 𝑥 = 𝐴 ↔ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
1514pm5.32da 579 . . . . . . . . . . 11 (Lim 𝐴 → ((𝑥𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1613, 15bitrid 283 . . . . . . . . . 10 (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1711, 16bitrid 283 . . . . . . . . 9 (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1817anbi2d 630 . . . . . . . 8 (Lim 𝐴 → ((𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
1910, 18bitrid 283 . . . . . . 7 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
2019exbidv 1919 . . . . . 6 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
219, 20bitrid 283 . . . . 5 (Lim 𝐴 → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
2221abbidv 2806 . . . 4 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
2322inteqd 4956 . . 3 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
248, 23eqtr2id 2788 . 2 (Lim 𝐴 {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))} = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
256, 24eqtrd 2775 1 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   cint 4951   ciin 4997  Ord word 6385  Oncon0 6386  Lim wlim 6387  cfv 6563  cardccrd 9973  cfccf 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-iota 6516  df-fun 6565  df-fv 6571  df-cf 9979
This theorem is referenced by:  cflim2  10301  cfss  10303  cfslb  10304
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