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Theorem cflm 10290
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 3501 . 2 (𝐴𝐵𝐴 ∈ V)
2 limsuc 7870 . . . . . . . . . . . . . . . . . 18 (Lim 𝐴 → (𝑣𝐴 ↔ suc 𝑣𝐴))
32biimpd 229 . . . . . . . . . . . . . . . . 17 (Lim 𝐴 → (𝑣𝐴 → suc 𝑣𝐴))
4 sseq1 4009 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = suc 𝑣 → (𝑧𝑤 ↔ suc 𝑣𝑤))
54rexbidv 3179 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc 𝑣 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝑦 suc 𝑣𝑤))
65rspcv 3618 . . . . . . . . . . . . . . . . . 18 (suc 𝑣𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∃𝑤𝑦 suc 𝑣𝑤))
7 sucssel 6479 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → (suc 𝑣𝑤𝑣𝑤))
87elv 3485 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑣𝑤𝑣𝑤)
98reximi 3084 . . . . . . . . . . . . . . . . . . 19 (∃𝑤𝑦 suc 𝑣𝑤 → ∃𝑤𝑦 𝑣𝑤)
10 eluni2 4911 . . . . . . . . . . . . . . . . . . 19 (𝑣 𝑦 ↔ ∃𝑤𝑦 𝑣𝑤)
119, 10sylibr 234 . . . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . . . 15 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∀𝑣𝐴 𝑣 𝑦))
15 dfss3 3972 . . . . . . . . . . . . . . 15 (𝐴 𝑦 ↔ ∀𝑣𝐴 𝑣 𝑦)
1614, 15imbitrrdi 252 . . . . . . . . . . . . . 14 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
1716adantr 480 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
18 uniss 4915 . . . . . . . . . . . . . . 15 (𝑦𝐴 𝑦 𝐴)
19 limuni 6445 . . . . . . . . . . . . . . . 16 (Lim 𝐴𝐴 = 𝐴)
2019sseq2d 4016 . . . . . . . . . . . . . . 15 (Lim 𝐴 → ( 𝑦𝐴 𝑦 𝐴))
2118, 20imbitrrid 246 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝑦𝐴 𝑦𝐴))
2221imp 406 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → 𝑦𝐴)
2317, 22jctird 526 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝐴 𝑦 𝑦𝐴)))
24 eqss 3999 . . . . . . . . . . . 12 (𝐴 = 𝑦 ↔ (𝐴 𝑦 𝑦𝐴))
2523, 24imbitrrdi 252 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 = 𝑦))
2625imdistanda 571 . . . . . . . . . 10 (Lim 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) → (𝑦𝐴𝐴 = 𝑦)))
2726anim2d 612 . . . . . . . . 9 (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2827eximdv 1917 . . . . . . . 8 (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2928ss2abdv 4066 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
30 intss 4969 . . . . . . 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 481 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
33 limelon 6448 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34 cfval 10287 . . . . . 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 4021 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴))
37 cfub 10289 . . . . 5 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
38 eqimss 4042 . . . . . . . . . 10 (𝐴 = 𝑦𝐴 𝑦)
3938anim2i 617 . . . . . . . . 9 ((𝑦𝐴𝐴 = 𝑦) → (𝑦𝐴𝐴 𝑦))
4039anim2i 617 . . . . . . . 8 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4140eximi 1835 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4241ss2abi 4067 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
43 intss 4969 . . . . . 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 3993 . . . 4 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))}
4636, 45jctil 519 . . 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 3999 . . 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 234 . 2 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
491, 48sylan 580 1 ((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951   cuni 4907   cint 4946  Oncon0 6384  Lim wlim 6385  suc csuc 6386  cfv 6561  cardccrd 9975  cfccf 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-card 9979  df-cf 9981
This theorem is referenced by:  gruina  10858
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