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Theorem cfub 9056
Description: An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfub (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem cfub
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 9054 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 dfss3 3585 . . . . . . . . 9 (𝐴 𝑦 ↔ ∀𝑧𝐴 𝑧 𝑦)
3 ssel 3589 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
4 onelon 5736 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑤𝐴) → 𝑤 ∈ On)
54ex 450 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → (𝑤𝐴𝑤 ∈ On))
63, 5sylan9r 689 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦𝑤 ∈ On))
7 onelss 5754 . . . . . . . . . . . . . . 15 (𝑤 ∈ On → (𝑧𝑤𝑧𝑤))
86, 7syl6 35 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦 → (𝑧𝑤𝑧𝑤)))
98imdistand 727 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑤𝑦𝑧𝑤) → (𝑤𝑦𝑧𝑤)))
109ancomsd 470 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑧𝑤𝑤𝑦) → (𝑤𝑦𝑧𝑤)))
1110eximdv 1844 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∃𝑤(𝑧𝑤𝑤𝑦) → ∃𝑤(𝑤𝑦𝑧𝑤)))
12 eluni 4430 . . . . . . . . . . 11 (𝑧 𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑦))
13 df-rex 2915 . . . . . . . . . . 11 (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤(𝑤𝑦𝑧𝑤))
1411, 12, 133imtr4g 285 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑧 𝑦 → ∃𝑤𝑦 𝑧𝑤))
1514ralimdv 2960 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∀𝑧𝐴 𝑧 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
162, 15syl5bi 232 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝐴 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716imdistanda 728 . . . . . . 7 (𝐴 ∈ On → ((𝑦𝐴𝐴 𝑦) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1817anim2d 588 . . . . . 6 (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918eximdv 1844 . . . . 5 (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2019ss2abdv 3667 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4489 . . . 4 ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
2220, 21syl 17 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
231, 22eqsstrd 3631 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
24 cff 9055 . . . . . 6 cf:On⟶On
2524fdmi 6039 . . . . 5 dom cf = On
2625eleq2i 2691 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6205 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
2826, 27sylnbir 321 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
29 0ss 3963 . . 3 ∅ ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
3028, 29syl6eqss 3647 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
3123, 30pm2.61i 176 1 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  wss 3567  c0 3907   cuni 4427   cint 4466  dom cdm 5104  Oncon0 5711  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-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:  cflm  9057  cf0  9058
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