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Theorem cfub 9659
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 9657 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 dfss3 3953 . . . . . . . . 9 (𝐴 𝑦 ↔ ∀𝑧𝐴 𝑧 𝑦)
3 ssel 3958 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
4 onelon 6209 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑤𝐴) → 𝑤 ∈ On)
54ex 413 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → (𝑤𝐴𝑤 ∈ On))
63, 5sylan9r 509 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦𝑤 ∈ On))
7 onelss 6226 . . . . . . . . . . . . . . 15 (𝑤 ∈ On → (𝑧𝑤𝑧𝑤))
86, 7syl6 35 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦 → (𝑧𝑤𝑧𝑤)))
98imdistand 571 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑤𝑦𝑧𝑤) → (𝑤𝑦𝑧𝑤)))
109ancomsd 466 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑧𝑤𝑤𝑦) → (𝑤𝑦𝑧𝑤)))
1110eximdv 1909 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∃𝑤(𝑧𝑤𝑤𝑦) → ∃𝑤(𝑤𝑦𝑧𝑤)))
12 eluni 4833 . . . . . . . . . . 11 (𝑧 𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑦))
13 df-rex 3141 . . . . . . . . . . 11 (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤(𝑤𝑦𝑧𝑤))
1411, 12, 133imtr4g 297 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑧 𝑦 → ∃𝑤𝑦 𝑧𝑤))
1514ralimdv 3175 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∀𝑧𝐴 𝑧 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
162, 15syl5bi 243 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝐴 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716imdistanda 572 . . . . . . 7 (𝐴 ∈ On → ((𝑦𝐴𝐴 𝑦) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1817anim2d 611 . . . . . 6 (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918eximdv 1909 . . . . 5 (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2019ss2abdv 4041 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4888 . . . 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 4002 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
24 cff 9658 . . . . . 6 cf:On⟶On
2524fdmi 6517 . . . . 5 dom cf = On
2625eleq2i 2901 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6693 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
2826, 27sylnbir 332 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
29 0ss 4347 . . 3 ∅ ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
3028, 29eqsstrdi 4018 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
3123, 30pm2.61i 183 1 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  wss 3933  c0 4288   cuni 4830   cint 4867  dom cdm 5548  Oncon0 6184  cfv 6348  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-card 9356  df-cf 9358
This theorem is referenced by:  cflm  9660  cf0  9661
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