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Theorem cfub 9669
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 9667 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 dfss3 3941 . . . . . . . . 9 (𝐴 𝑦 ↔ ∀𝑧𝐴 𝑧 𝑦)
3 ssel 3946 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
4 onelon 6203 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑤𝐴) → 𝑤 ∈ On)
54ex 416 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → (𝑤𝐴𝑤 ∈ On))
63, 5sylan9r 512 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦𝑤 ∈ On))
7 onelss 6220 . . . . . . . . . . . . . . 15 (𝑤 ∈ On → (𝑧𝑤𝑧𝑤))
86, 7syl6 35 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦 → (𝑧𝑤𝑧𝑤)))
98imdistand 574 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑤𝑦𝑧𝑤) → (𝑤𝑦𝑧𝑤)))
109ancomsd 469 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑧𝑤𝑤𝑦) → (𝑤𝑦𝑧𝑤)))
1110eximdv 1919 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∃𝑤(𝑧𝑤𝑤𝑦) → ∃𝑤(𝑤𝑦𝑧𝑤)))
12 eluni 4827 . . . . . . . . . . 11 (𝑧 𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑦))
13 df-rex 3139 . . . . . . . . . . 11 (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤(𝑤𝑦𝑧𝑤))
1411, 12, 133imtr4g 299 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑧 𝑦 → ∃𝑤𝑦 𝑧𝑤))
1514ralimdv 3173 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∀𝑧𝐴 𝑧 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
162, 15syl5bi 245 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝐴 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716imdistanda 575 . . . . . . 7 (𝐴 ∈ On → ((𝑦𝐴𝐴 𝑦) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1817anim2d 614 . . . . . 6 (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918eximdv 1919 . . . . 5 (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2019ss2abdv 4030 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4883 . . . 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 3991 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
24 cff 9668 . . . . . 6 cf:On⟶On
2524fdmi 6514 . . . . 5 dom cf = On
2625eleq2i 2907 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6691 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
2826, 27sylnbir 334 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
29 0ss 4333 . . 3 ∅ ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
3028, 29eqsstrdi 4007 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
3123, 30pm2.61i 185 1 (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wrex 3134  wss 3919  c0 4276   cuni 4824   cint 4862  dom cdm 5542  Oncon0 6178  cfv 6343  cardccrd 9361  cfccf 9363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-card 9365  df-cf 9367
This theorem is referenced by:  cflm  9670  cf0  9671
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