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Theorem cflim3 9677
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 6222 . . . 4 (Lim 𝐴 → Ord 𝐴)
2 cflim3.1 . . . . 5 𝐴 ∈ V
32elon 6172 . . . 4 (𝐴 ∈ On ↔ Ord 𝐴)
41, 3sylibr 237 . . 3 (Lim 𝐴𝐴 ∈ On)
5 cfval 9662 . . 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 6662 . . . 4 (card‘𝑥) ∈ V
87dfiin2 4924 . . 3 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥) = {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9 df-rex 3115 . . . . . 6 (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10 ancom 464 . . . . . . . 8 ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}))
11 rabid 3334 . . . . . . . . . 10 (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴))
12 velpw 4505 . . . . . . . . . . . 12 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1312anbi1i 626 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 𝑥 = 𝐴))
14 coflim 9676 . . . . . . . . . . . 12 ((Lim 𝐴𝑥𝐴) → ( 𝑥 = 𝐴 ↔ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))
1514pm5.32da 582 . . . . . . . . . . 11 (Lim 𝐴 → ((𝑥𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1613, 15syl5bb 286 . . . . . . . . . 10 (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴) ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1711, 16syl5bb 286 . . . . . . . . 9 (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ↔ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤)))
1817anbi2d 631 . . . . . . . 8 (Lim 𝐴 → ((𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
1910, 18syl5bb 286 . . . . . . 7 (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
2019exbidv 1922 . . . . . 6 (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
219, 20syl5bb 286 . . . . 5 (Lim 𝐴 → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))))
2221abbidv 2865 . . . 4 (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
2322inteqd 4846 . . 3 (Lim 𝐴 {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))})
248, 23syl5req 2849 . 2 (Lim 𝐴 {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥𝐴 ∧ ∀𝑧𝐴𝑤𝑥 𝑧𝑤))} = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
256, 24eqtrd 2836 1 (Lim 𝐴 → (cf‘𝐴) = 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 𝑥 = 𝐴} (card‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  𝒫 cpw 4500   cuni 4803   cint 4841   ciin 4885  Ord word 6162  Oncon0 6163  Lim wlim 6164  cfv 6328  cardccrd 9352  cfccf 9354
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-ord 6166  df-on 6167  df-lim 6168  df-iota 6287  df-fun 6330  df-fv 6336  df-cf 9358
This theorem is referenced by:  cflim2  9678  cfss  9680  cfslb  9681
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