MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cflecard Structured version   Visualization version   GIF version

Theorem cflecard 10197
Description: Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cflecard (cfβ€˜π΄) βŠ† (cardβ€˜π΄)

Proof of Theorem cflecard
Dummy variables π‘₯ 𝑦 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 df-sn 4591 . . . . . 6 {(cardβ€˜π΄)} = {π‘₯ ∣ π‘₯ = (cardβ€˜π΄)}
3 ssid 3970 . . . . . . . . 9 𝐴 βŠ† 𝐴
4 ssid 3970 . . . . . . . . . . 11 𝑧 βŠ† 𝑧
5 sseq2 3974 . . . . . . . . . . . 12 (𝑀 = 𝑧 β†’ (𝑧 βŠ† 𝑀 ↔ 𝑧 βŠ† 𝑧))
65rspcev 3583 . . . . . . . . . . 11 ((𝑧 ∈ 𝐴 ∧ 𝑧 βŠ† 𝑧) β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
74, 6mpan2 690 . . . . . . . . . 10 (𝑧 ∈ 𝐴 β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
87rgen 3063 . . . . . . . . 9 βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀
93, 8pm3.2i 472 . . . . . . . 8 (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
10 fveq2 6846 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (cardβ€˜π‘¦) = (cardβ€˜π΄))
1110eqeq2d 2744 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ π‘₯ = (cardβ€˜π΄)))
12 sseq1 3973 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (𝑦 βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
13 rexeq 3309 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
1413ralbidv 3171 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
1512, 14anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ ((𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)))
1611, 15anbi12d 632 . . . . . . . . 9 (𝑦 = 𝐴 β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (π‘₯ = (cardβ€˜π΄) ∧ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))))
1716spcegv 3558 . . . . . . . 8 (𝐴 ∈ On β†’ ((π‘₯ = (cardβ€˜π΄) ∧ (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)) β†’ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
189, 17mpan2i 696 . . . . . . 7 (𝐴 ∈ On β†’ (π‘₯ = (cardβ€˜π΄) β†’ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
1918ss2abdv 4024 . . . . . 6 (𝐴 ∈ On β†’ {π‘₯ ∣ π‘₯ = (cardβ€˜π΄)} βŠ† {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
202, 19eqsstrid 3996 . . . . 5 (𝐴 ∈ On β†’ {(cardβ€˜π΄)} βŠ† {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
21 intss 4934 . . . . 5 ({(cardβ€˜π΄)} 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† ∩ {(cardβ€˜π΄)})
2220, 21syl 17 . . . 4 (𝐴 ∈ On β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† ∩ {(cardβ€˜π΄)})
23 fvex 6859 . . . . 5 (cardβ€˜π΄) ∈ V
2423intsn 4951 . . . 4 ∩ {(cardβ€˜π΄)} = (cardβ€˜π΄)
2522, 24sseqtrdi 3998 . . 3 (𝐴 ∈ On β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} βŠ† (cardβ€˜π΄))
261, 25eqsstrd 3986 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† (cardβ€˜π΄))
27 cff 10192 . . . . . 6 cf:On⟢On
2827fdmi 6684 . . . . 5 dom cf = On
2928eleq2i 2826 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6881 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
3129, 30sylnbir 331 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
32 0ss 4360 . . 3 βˆ… βŠ† (cardβ€˜π΄)
3331, 32eqsstrdi 4002 . 2 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† (cardβ€˜π΄))
3426, 33pm2.61i 182 1 (cfβ€˜π΄) βŠ† (cardβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914  βˆ…c0 4286  {csn 4590  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:  cfle  10198
  Copyright terms: Public domain W3C validator