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Theorem cflm 9671
Description: Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
cflm ((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem cflm
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3512 . 2 (𝐴𝐵𝐴 ∈ V)
2 limsuc 7563 . . . . . . . . . . . . . . . . . 18 (Lim 𝐴 → (𝑣𝐴 ↔ suc 𝑣𝐴))
32biimpd 231 . . . . . . . . . . . . . . . . 17 (Lim 𝐴 → (𝑣𝐴 → suc 𝑣𝐴))
4 sseq1 3991 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = suc 𝑣 → (𝑧𝑤 ↔ suc 𝑣𝑤))
54rexbidv 3297 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc 𝑣 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝑦 suc 𝑣𝑤))
65rspcv 3617 . . . . . . . . . . . . . . . . . 18 (suc 𝑣𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∃𝑤𝑦 suc 𝑣𝑤))
7 sucssel 6282 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → (suc 𝑣𝑤𝑣𝑤))
87elv 3499 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑣𝑤𝑣𝑤)
98reximi 3243 . . . . . . . . . . . . . . . . . . 19 (∃𝑤𝑦 suc 𝑣𝑤 → ∃𝑤𝑦 𝑣𝑤)
10 eluni2 4841 . . . . . . . . . . . . . . . . . . 19 (𝑣 𝑦 ↔ ∃𝑤𝑦 𝑣𝑤)
119, 10sylibr 236 . . . . . . . . . . . . . . . . . 18 (∃𝑤𝑦 suc 𝑣𝑤𝑣 𝑦)
126, 11syl6com 37 . . . . . . . . . . . . . . . . 17 (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (suc 𝑣𝐴𝑣 𝑦))
133, 12syl9 77 . . . . . . . . . . . . . . . 16 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝑣𝐴𝑣 𝑦)))
1413ralrimdv 3188 . . . . . . . . . . . . . . 15 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∀𝑣𝐴 𝑣 𝑦))
15 dfss3 3955 . . . . . . . . . . . . . . 15 (𝐴 𝑦 ↔ ∀𝑣𝐴 𝑣 𝑦)
1614, 15syl6ibr 254 . . . . . . . . . . . . . 14 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
1716adantr 483 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
18 uniss 4845 . . . . . . . . . . . . . . 15 (𝑦𝐴 𝑦 𝐴)
19 limuni 6250 . . . . . . . . . . . . . . . 16 (Lim 𝐴𝐴 = 𝐴)
2019sseq2d 3998 . . . . . . . . . . . . . . 15 (Lim 𝐴 → ( 𝑦𝐴 𝑦 𝐴))
2118, 20syl5ibr 248 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝑦𝐴 𝑦𝐴))
2221imp 409 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → 𝑦𝐴)
2317, 22jctird 529 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝐴 𝑦 𝑦𝐴)))
24 eqss 3981 . . . . . . . . . . . 12 (𝐴 = 𝑦 ↔ (𝐴 𝑦 𝑦𝐴))
2523, 24syl6ibr 254 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 = 𝑦))
2625imdistanda 574 . . . . . . . . . 10 (Lim 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) → (𝑦𝐴𝐴 = 𝑦)))
2726anim2d 613 . . . . . . . . 9 (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2827eximdv 1914 . . . . . . . 8 (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2928ss2abdv 4043 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
30 intss 4896 . . . . . . 7 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3129, 30syl 17 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3231adantl 484 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
33 limelon 6253 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34 cfval 9668 . . . . . 6 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3533, 34syl 17 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3632, 35sseqtrrd 4007 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴))
37 cfub 9670 . . . . 5 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
38 eqimss 4022 . . . . . . . . . 10 (𝐴 = 𝑦𝐴 𝑦)
3938anim2i 618 . . . . . . . . 9 ((𝑦𝐴𝐴 = 𝑦) → (𝑦𝐴𝐴 𝑦))
4039anim2i 618 . . . . . . . 8 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4140eximi 1831 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4241ss2abi 4042 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
43 intss 4896 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
4442, 43ax-mp 5 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))}
4537, 44sstri 3975 . . . 4 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))}
4636, 45jctil 522 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → ((cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴)))
47 eqss 3981 . . 3 ((cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ↔ ((cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴)))
4846, 47sylibr 236 . 2 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
491, 48sylan 582 1 ((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wex 1776  wcel 2110  {cab 2799  wral 3138  wrex 3139  Vcvv 3494  wss 3935   cuni 4837   cint 4875  Oncon0 6190  Lim wlim 6191  suc csuc 6192  cfv 6354  cardccrd 9363  cfccf 9365
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-card 9367  df-cf 9369
This theorem is referenced by:  gruina  10239
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