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Theorem cfub 10193
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 10191 . . 3 (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
2 dfss3 3936 . . . . . . . . 9 (𝐴 βŠ† βˆͺ 𝑦 ↔ βˆ€π‘§ ∈ 𝐴 𝑧 ∈ βˆͺ 𝑦)
3 ssel 3941 . . . . . . . . . . . . . . . 16 (𝑦 βŠ† 𝐴 β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 ∈ 𝐴))
4 onelon 6346 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝑀 ∈ 𝐴) β†’ 𝑀 ∈ On)
54ex 414 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On β†’ (𝑀 ∈ 𝐴 β†’ 𝑀 ∈ On))
63, 5sylan9r 510 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 ∈ On))
7 onelss 6363 . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ ((𝑀 ∈ 𝑦 ∧ 𝑧 ∈ 𝑀) β†’ (𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
109ancomsd 467 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ ((𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝑦) β†’ (𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
1110eximdv 1921 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (βˆƒπ‘€(𝑧 ∈ 𝑀 ∧ 𝑀 ∈ 𝑦) β†’ βˆƒπ‘€(𝑀 ∈ 𝑦 ∧ 𝑧 βŠ† 𝑀)))
12 eluni 4872 . . . . . . . . . . 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 296 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦 βŠ† 𝐴) β†’ (𝑧 ∈ βˆͺ 𝑦 β†’ βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
1514ralimdv 3163 . . . . . . . . 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 573 . . . . . . 7 (𝐴 ∈ On β†’ ((𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦) β†’ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
1817anim2d 613 . . . . . 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 1921 . . . . 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 4024 . . . 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 4934 . . . 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 3986 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))})
24 cff 10192 . . . . . 6 cf:On⟢On
2524fdmi 6684 . . . . 5 dom cf = On
2625eleq2i 2826 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27 ndmfv 6881 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
2826, 27sylnbir 331 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
29 0ss 4360 . . 3 βˆ… βŠ† ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))}
3028, 29eqsstrdi 4002 . 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914  βˆ…c0 4286  βˆͺ cuni 4869  βˆ© cint 4911  dom cdm 5637  Oncon0 6321  β€˜cfv 6500  cardccrd 9879  cfccf 9881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-card 9883  df-cf 9885
This theorem is referenced by:  cflm  10194  cf0  10195
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