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Theorem cflm 9057
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 3207 . 2 (𝐴𝐵𝐴 ∈ V)
2 limsuc 7034 . . . . . . . . . . . . . . . . . 18 (Lim 𝐴 → (𝑣𝐴 ↔ suc 𝑣𝐴))
32biimpd 219 . . . . . . . . . . . . . . . . 17 (Lim 𝐴 → (𝑣𝐴 → suc 𝑣𝐴))
4 sseq1 3618 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = suc 𝑣 → (𝑧𝑤 ↔ suc 𝑣𝑤))
54rexbidv 3048 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc 𝑣 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝑦 suc 𝑣𝑤))
65rspcv 3300 . . . . . . . . . . . . . . . . . 18 (suc 𝑣𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∃𝑤𝑦 suc 𝑣𝑤))
7 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
8 sucssel 5807 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → (suc 𝑣𝑤𝑣𝑤))
97, 8ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑣𝑤𝑣𝑤)
109reximi 3008 . . . . . . . . . . . . . . . . . . 19 (∃𝑤𝑦 suc 𝑣𝑤 → ∃𝑤𝑦 𝑣𝑤)
11 eluni2 4431 . . . . . . . . . . . . . . . . . . 19 (𝑣 𝑦 ↔ ∃𝑤𝑦 𝑣𝑤)
1210, 11sylibr 224 . . . . . . . . . . . . . . . . . 18 (∃𝑤𝑦 suc 𝑣𝑤𝑣 𝑦)
136, 12syl6com 37 . . . . . . . . . . . . . . . . 17 (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (suc 𝑣𝐴𝑣 𝑦))
143, 13syl9 77 . . . . . . . . . . . . . . . 16 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝑣𝐴𝑣 𝑦)))
1514ralrimdv 2965 . . . . . . . . . . . . . . 15 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∀𝑣𝐴 𝑣 𝑦))
16 dfss3 3585 . . . . . . . . . . . . . . 15 (𝐴 𝑦 ↔ ∀𝑣𝐴 𝑣 𝑦)
1715, 16syl6ibr 242 . . . . . . . . . . . . . 14 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
1817adantr 481 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
19 uniss 4449 . . . . . . . . . . . . . . 15 (𝑦𝐴 𝑦 𝐴)
20 limuni 5773 . . . . . . . . . . . . . . . 16 (Lim 𝐴𝐴 = 𝐴)
2120sseq2d 3625 . . . . . . . . . . . . . . 15 (Lim 𝐴 → ( 𝑦𝐴 𝑦 𝐴))
2219, 21syl5ibr 236 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝑦𝐴 𝑦𝐴))
2322imp 445 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → 𝑦𝐴)
2418, 23jctird 566 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝐴 𝑦 𝑦𝐴)))
25 eqss 3610 . . . . . . . . . . . 12 (𝐴 = 𝑦 ↔ (𝐴 𝑦 𝑦𝐴))
2624, 25syl6ibr 242 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 = 𝑦))
2726imdistanda 728 . . . . . . . . . 10 (Lim 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) → (𝑦𝐴𝐴 = 𝑦)))
2827anim2d 588 . . . . . . . . 9 (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2928eximdv 1844 . . . . . . . 8 (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
3029ss2abdv 3667 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
31 intss 4489 . . . . . . 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))})
3230, 31syl 17 . . . . . 6 (Lim 𝐴 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3332adantl 482 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
34 limelon 5776 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
35 cfval 9054 . . . . . 6 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3634, 35syl 17 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
3733, 36sseqtr4d 3634 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴))
38 cfub 9056 . . . . 5 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
39 eqimss 3649 . . . . . . . . . 10 (𝐴 = 𝑦𝐴 𝑦)
4039anim2i 592 . . . . . . . . 9 ((𝑦𝐴𝐴 = 𝑦) → (𝑦𝐴𝐴 𝑦))
4140anim2i 592 . . . . . . . 8 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4241eximi 1760 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4342ss2abi 3666 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
44 intss 4489 . . . . . 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))})
4543, 44ax-mp 5 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))}
4638, 45sstri 3604 . . . 4 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))}
4737, 46jctil 559 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → ((cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴)))
48 eqss 3610 . . 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)))
4947, 48sylibr 224 . 2 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
501, 49sylan 488 1 ((𝐴𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  Vcvv 3195  wss 3567   cuni 4427   cint 4466  Oncon0 5711  Lim wlim 5712  suc csuc 5713  cfv 5876  cardccrd 8746  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-card 8750  df-cf 8752
This theorem is referenced by:  gruina  9625
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