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Theorem cfub 10243
Description: An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfub (cfβ€˜π΄) βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))}
Distinct variable group:   π‘₯,𝑦,𝐴

Proof of Theorem cfub
Dummy variables 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 10241 . . 3 (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
2 dfss3 3970 . . . . . . . . 9 (𝐴 βŠ† βˆͺ 𝑦 ↔ βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑦)
3 ssel 3975 . . . . . . . . . . . . . . . 16 (𝑦 βŠ† 𝐴 β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 ∈ 𝐴))
4 onelon 6389 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑀 ∈ 𝐴) β†’ 𝑀 ∈ On)
54ex 413 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 ∈ On))
63, 5sylan9r 509 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 ∈ On))
7 onelss 6406 . . . . . . . . . . . . . . 15 (𝑀 ∈ On β†’ (𝑧 ∈ 𝑀 β†’ 𝑧 βŠ† 𝑀))
86, 7syl6 35 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝑀 ∈ 𝑦 β†’ (𝑧 ∈ 𝑀 β†’ 𝑧 βŠ† 𝑀)))
98imdistand 571 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ ((𝑀 ∈ 𝑦 ∧ 𝑧 ∈ 𝑀) β†’ (𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
109ancomsd 466 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ ((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝑦) β†’ (𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
1110eximdv 1920 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝑦) β†’ βˆƒπ‘€(𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
12 eluni 4911 . . . . . . . . . . 11 (𝑧 ∈ βˆͺ 𝑦 ↔ βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝑦))
13 df-rex 3071 . . . . . . . . . . 11 (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€(𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀))
1411, 12, 133imtr4g 295 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝑧 ∈ βˆͺ 𝑦 β†’ βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
1514ralimdv 3169 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑦 β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
162, 15biimtrid 241 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝐴 βŠ† βˆͺ 𝑦 β†’ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
1716imdistanda 572 . . . . . . 7 (𝐴 ∈ On β†’ ((𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦) β†’ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
1817anim2d 612 . . . . . 6 (𝐴 ∈ On β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦)) β†’ (π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
1918eximdv 1920 . . . . 5 (𝐴 ∈ On β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦)) β†’ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
2019ss2abdv 4060 . . . 4 (𝐴 ∈ On β†’ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))} βŠ† {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
21 intss 4973 . . . 4 ({π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))} βŠ† {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))})
2220, 21syl 17 . . 3 (𝐴 ∈ On β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))})
231, 22eqsstrd 4020 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))})
24 cff 10242 . . . . . 6 cf:On⟢On
2524fdmi 6729 . . . . 5 dom cf = On
2625eleq2i 2825 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6926 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
2826, 27sylnbir 330 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
29 0ss 4396 . . 3 βˆ… βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))}
3028, 29eqsstrdi 4036 . 2 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))})
3123, 30pm2.61i 182 1 (cfβ€˜π΄) βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  βˆ…c0 4322  βˆͺ cuni 4908  βˆ© cint 4950  dom cdm 5676  Oncon0 6364  β€˜cfv 6543  cardccrd 9929  cfccf 9931
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-card 9933  df-cf 9935
This theorem is referenced by:  cflm  10244  cf0  10245
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