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Theorem cfub 8954
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 8952 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 dfss3 3558 . . . . . . . . 9 (𝐴 𝑦 ↔ ∀𝑧𝐴 𝑧 𝑦)
3 ssel 3562 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
4 onelon 5665 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑤𝐴) → 𝑤 ∈ On)
54ex 449 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → (𝑤𝐴𝑤 ∈ On))
63, 5sylan9r 688 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦𝑤 ∈ On))
7 onelss 5683 . . . . . . . . . . . . . . 15 (𝑤 ∈ On → (𝑧𝑤𝑧𝑤))
86, 7syl6 34 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦 → (𝑧𝑤𝑧𝑤)))
98imdistand 724 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑤𝑦𝑧𝑤) → (𝑤𝑦𝑧𝑤)))
109ancomsd 469 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑧𝑤𝑤𝑦) → (𝑤𝑦𝑧𝑤)))
1110eximdv 1833 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∃𝑤(𝑧𝑤𝑤𝑦) → ∃𝑤(𝑤𝑦𝑧𝑤)))
12 eluni 4375 . . . . . . . . . . 11 (𝑧 𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑦))
13 df-rex 2902 . . . . . . . . . . 11 (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤(𝑤𝑦𝑧𝑤))
1411, 12, 133imtr4g 284 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑧 𝑦 → ∃𝑤𝑦 𝑧𝑤))
1514ralimdv 2946 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∀𝑧𝐴 𝑧 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
162, 15syl5bi 231 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝐴 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716imdistanda 725 . . . . . . 7 (𝐴 ∈ On → ((𝑦𝐴𝐴 𝑦) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1817anim2d 587 . . . . . 6 (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918eximdv 1833 . . . . 5 (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2019ss2abdv 3638 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4433 . . . 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 3602 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
24 cff 8953 . . . . . 6 cf:On⟶On
2524fdmi 5965 . . . . 5 dom cf = On
2625eleq2i 2680 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6128 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
2826, 27sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
29 0ss 3924 . . 3 ∅ ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
3028, 29syl6eqss 3618 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
3123, 30pm2.61i 175 1 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  wss 3540  c0 3874   cuni 4372   cint 4410  dom cdm 5038  Oncon0 5640  cfv 5804  cardccrd 8644  cfccf 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-card 8648  df-cf 8650
This theorem is referenced by:  cflm  8955  cf0  8956
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