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Theorem cflm 9016
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 3198 . 2 (𝐴𝐵𝐴 ∈ V)
2 limsuc 6996 . . . . . . . . . . . . . . . . . 18 (Lim 𝐴 → (𝑣𝐴 ↔ suc 𝑣𝐴))
32biimpd 219 . . . . . . . . . . . . . . . . 17 (Lim 𝐴 → (𝑣𝐴 → suc 𝑣𝐴))
4 sseq1 3605 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = suc 𝑣 → (𝑧𝑤 ↔ suc 𝑣𝑤))
54rexbidv 3045 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc 𝑣 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝑦 suc 𝑣𝑤))
65rspcv 3291 . . . . . . . . . . . . . . . . . 18 (suc 𝑣𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∃𝑤𝑦 suc 𝑣𝑤))
7 vex 3189 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
8 sucssel 5778 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → (suc 𝑣𝑤𝑣𝑤))
97, 8ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑣𝑤𝑣𝑤)
109reximi 3005 . . . . . . . . . . . . . . . . . . 19 (∃𝑤𝑦 suc 𝑣𝑤 → ∃𝑤𝑦 𝑣𝑤)
11 eluni2 4406 . . . . . . . . . . . . . . . . . . 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 2962 . . . . . . . . . . . . . . 15 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∀𝑣𝐴 𝑣 𝑦))
16 dfss3 3573 . . . . . . . . . . . . . . 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 4424 . . . . . . . . . . . . . . 15 (𝑦𝐴 𝑦 𝐴)
20 limuni 5744 . . . . . . . . . . . . . . . 16 (Lim 𝐴𝐴 = 𝐴)
2120sseq2d 3612 . . . . . . . . . . . . . . 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 3598 . . . . . . . . . . . 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 1843 . . . . . . . 8 (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
3029ss2abdv 3654 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
31 intss 4463 . . . . . . 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 5747 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
35 cfval 9013 . . . . . 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 3621 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ (cf‘𝐴))
38 cfub 9015 . . . . 5 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
39 eqimss 3636 . . . . . . . . . 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 1759 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4342ss2abi 3653 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
44 intss 4463 . . . . . 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 3592 . . . 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 3598 . . 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 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  wss 3555   cuni 4402   cint 4440  Oncon0 5682  Lim wlim 5683  suc csuc 5684  cfv 5847  cardccrd 8705  cfccf 8707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-card 8709  df-cf 8711
This theorem is referenced by:  gruina  9584
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