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Theorem cfub 9273
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 9271 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 dfss3 3741 . . . . . . . . 9 (𝐴 𝑦 ↔ ∀𝑧𝐴 𝑧 𝑦)
3 ssel 3746 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → (𝑤𝑦𝑤𝐴))
4 onelon 5891 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑤𝐴) → 𝑤 ∈ On)
54ex 397 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → (𝑤𝐴𝑤 ∈ On))
63, 5sylan9r 498 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦𝑤 ∈ On))
7 onelss 5909 . . . . . . . . . . . . . . 15 (𝑤 ∈ On → (𝑧𝑤𝑧𝑤))
86, 7syl6 35 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑤𝑦 → (𝑧𝑤𝑧𝑤)))
98imdistand 560 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑤𝑦𝑧𝑤) → (𝑤𝑦𝑧𝑤)))
109ancomsd 456 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((𝑧𝑤𝑤𝑦) → (𝑤𝑦𝑧𝑤)))
1110eximdv 1998 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∃𝑤(𝑧𝑤𝑤𝑦) → ∃𝑤(𝑤𝑦𝑧𝑤)))
12 eluni 4577 . . . . . . . . . . 11 (𝑧 𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑦))
13 df-rex 3067 . . . . . . . . . . 11 (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤(𝑤𝑦𝑧𝑤))
1411, 12, 133imtr4g 285 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝑧 𝑦 → ∃𝑤𝑦 𝑧𝑤))
1514ralimdv 3112 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → (∀𝑧𝐴 𝑧 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
162, 15syl5bi 232 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦𝐴) → (𝐴 𝑦 → ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
1716imdistanda 561 . . . . . . 7 (𝐴 ∈ On → ((𝑦𝐴𝐴 𝑦) → (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
1817anim2d 599 . . . . . 6 (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918eximdv 1998 . . . . 5 (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
2019ss2abdv 3824 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4632 . . . 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 3788 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))})
24 cff 9272 . . . . . 6 cf:On⟶On
2524fdmi 6192 . . . . 5 dom cf = On
2625eleq2i 2842 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6359 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
2826, 27sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
29 0ss 4116 . . 3 ∅ ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴𝐴 𝑦))}
3028, 29syl6eqss 3804 . 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 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  wss 3723  c0 4063   cuni 4574   cint 4611  dom cdm 5249  Oncon0 5866  cfv 6031  cardccrd 8961  cfccf 8963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-card 8965  df-cf 8967
This theorem is referenced by:  cflm  9274  cf0  9275
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