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Theorem cflm 10274
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 3490 . 2 (𝐴𝐵𝐴 ∈ V)
2 limsuc 7853 . . . . . . . . . . . . . . . . . 18 (Lim 𝐴 → (𝑣𝐴 ↔ suc 𝑣𝐴))
32biimpd 228 . . . . . . . . . . . . . . . . 17 (Lim 𝐴 → (𝑣𝐴 → suc 𝑣𝐴))
4 sseq1 4005 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = suc 𝑣 → (𝑧𝑤 ↔ suc 𝑣𝑤))
54rexbidv 3175 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc 𝑣 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝑦 suc 𝑣𝑤))
65rspcv 3605 . . . . . . . . . . . . . . . . . 18 (suc 𝑣𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∃𝑤𝑦 suc 𝑣𝑤))
7 sucssel 6464 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → (suc 𝑣𝑤𝑣𝑤))
87elv 3477 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑣𝑤𝑣𝑤)
98reximi 3081 . . . . . . . . . . . . . . . . . . 19 (∃𝑤𝑦 suc 𝑣𝑤 → ∃𝑤𝑦 𝑣𝑤)
10 eluni2 4912 . . . . . . . . . . . . . . . . . . 19 (𝑣 𝑦 ↔ ∃𝑤𝑦 𝑣𝑤)
119, 10sylibr 233 . . . . . . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . 15 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → ∀𝑣𝐴 𝑣 𝑦))
15 dfss3 3968 . . . . . . . . . . . . . . 15 (𝐴 𝑦 ↔ ∀𝑣𝐴 𝑣 𝑦)
1614, 15imbitrrdi 251 . . . . . . . . . . . . . 14 (Lim 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
1716adantr 480 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 𝑦))
18 uniss 4916 . . . . . . . . . . . . . . 15 (𝑦𝐴 𝑦 𝐴)
19 limuni 6430 . . . . . . . . . . . . . . . 16 (Lim 𝐴𝐴 = 𝐴)
2019sseq2d 4012 . . . . . . . . . . . . . . 15 (Lim 𝐴 → ( 𝑦𝐴 𝑦 𝐴))
2118, 20imbitrrid 245 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝑦𝐴 𝑦𝐴))
2221imp 406 . . . . . . . . . . . . 13 ((Lim 𝐴𝑦𝐴) → 𝑦𝐴)
2317, 22jctird 526 . . . . . . . . . . . 12 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 → (𝐴 𝑦 𝑦𝐴)))
24 eqss 3995 . . . . . . . . . . . 12 (𝐴 = 𝑦 ↔ (𝐴 𝑦 𝑦𝐴))
2523, 24imbitrrdi 251 . . . . . . . . . . 11 ((Lim 𝐴𝑦𝐴) → (∀𝑧𝐴𝑤𝑦 𝑧𝑤𝐴 = 𝑦))
2625imdistanda 571 . . . . . . . . . 10 (Lim 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) → (𝑦𝐴𝐴 = 𝑦)))
2726anim2d 611 . . . . . . . . 9 (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2827eximdv 1913 . . . . . . . 8 (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))))
2928ss2abdv 4058 . . . . . . 7 (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
30 intss 4972 . . . . . . 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 6433 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34 cfval 10271 . . . . . 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 10273 . . . . 5 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
38 eqimss 4038 . . . . . . . . . 10 (𝐴 = 𝑦𝐴 𝑦)
3938anim2i 616 . . . . . . . . 9 ((𝑦𝐴𝐴 = 𝑦) → (𝑦𝐴𝐴 𝑦))
4039anim2i 616 . . . . . . . 8 ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4140eximi 1830 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)))
4241ss2abi 4061 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
43 intss 4972 . . . . . 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 3989 . . . 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 3995 . . 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 233 . 2 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 = 𝑦))})
491, 48sylan 579 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 1534  wex 1774  wcel 2099  {cab 2705  wral 3058  wrex 3067  Vcvv 3471  wss 3947   cuni 4908   cint 4949  Oncon0 6369  Lim wlim 6370  suc csuc 6371  cfv 6548  cardccrd 9959  cfccf 9961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-card 9963  df-cf 9965
This theorem is referenced by:  gruina  10842
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