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Theorem cflecard 10247
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 10241 . . 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 4629 . . . . . 6 {(cardβ€˜π΄)} = {π‘₯ ∣ π‘₯ = (cardβ€˜π΄)}
3 ssid 4004 . . . . . . . . 9 𝐴 βŠ† 𝐴
4 ssid 4004 . . . . . . . . . . 11 𝑧 βŠ† 𝑧
5 sseq2 4008 . . . . . . . . . . . 12 (𝑀 = 𝑧 β†’ (𝑧 βŠ† 𝑀 ↔ 𝑧 βŠ† 𝑧))
65rspcev 3612 . . . . . . . . . . 11 ((𝑧 ∈ 𝐴 ∧ 𝑧 βŠ† 𝑧) β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
74, 6mpan2 689 . . . . . . . . . 10 (𝑧 ∈ 𝐴 β†’ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
87rgen 3063 . . . . . . . . 9 βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀
93, 8pm3.2i 471 . . . . . . . 8 (𝐴 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀)
10 fveq2 6891 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (cardβ€˜π‘¦) = (cardβ€˜π΄))
1110eqeq2d 2743 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ (π‘₯ = (cardβ€˜π‘¦) ↔ π‘₯ = (cardβ€˜π΄)))
12 sseq1 4007 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (𝑦 βŠ† 𝐴 ↔ 𝐴 βŠ† 𝐴))
13 rexeq 3321 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ (βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
1413ralbidv 3177 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝐴 𝑧 βŠ† 𝑀))
1512, 14anbi12d 631 . . . . . . . . . 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 631 . . . . . . . . 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 3587 . . . . . . . 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 695 . . . . . . 7 (𝐴 ∈ On β†’ (π‘₯ = (cardβ€˜π΄) β†’ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
1918ss2abdv 4060 . . . . . 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 4030 . . . . 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 4973 . . . . 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 6904 . . . . 5 (cardβ€˜π΄) ∈ V
2423intsn 4990 . . . 4 ∩ {(cardβ€˜π΄)} = (cardβ€˜π΄)
2522, 24sseqtrdi 4032 . . 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 4020 . 2 (𝐴 ∈ On β†’ (cfβ€˜π΄) βŠ† (cardβ€˜π΄))
27 cff 10242 . . . . . 6 cf:On⟢On
2827fdmi 6729 . . . . 5 dom cf = On
2928eleq2i 2825 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6926 . . . 4 (Β¬ 𝐴 ∈ dom cf β†’ (cfβ€˜π΄) = βˆ…)
3129, 30sylnbir 330 . . 3 (Β¬ 𝐴 ∈ On β†’ (cfβ€˜π΄) = βˆ…)
32 0ss 4396 . . 3 βˆ… βŠ† (cardβ€˜π΄)
3331, 32eqsstrdi 4036 . 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  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:  cfle  10248
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