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Theorem cflecard 10230
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 (cf‘𝐴) ⊆ (card‘𝐴)

Proof of Theorem cflecard
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 10224 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 df-sn 4623 . . . . . 6 {(card‘𝐴)} = {𝑥𝑥 = (card‘𝐴)}
3 ssid 4000 . . . . . . . . 9 𝐴𝐴
4 ssid 4000 . . . . . . . . . . 11 𝑧𝑧
5 sseq2 4004 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
65rspcev 3609 . . . . . . . . . . 11 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
74, 6mpan2 689 . . . . . . . . . 10 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
87rgen 3062 . . . . . . . . 9 𝑧𝐴𝑤𝐴 𝑧𝑤
93, 8pm3.2i 471 . . . . . . . 8 (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
10 fveq2 6878 . . . . . . . . . . 11 (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
1110eqeq2d 2742 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12 sseq1 4003 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
13 rexeq 3320 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
1413ralbidv 3176 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
1512, 14anbi12d 631 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1611, 15anbi12d 631 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))))
1716spcegv 3584 . . . . . . . 8 (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
189, 17mpan2i 695 . . . . . . 7 (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918ss2abdv 4056 . . . . . 6 (𝐴 ∈ On → {𝑥𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
202, 19eqsstrid 4026 . . . . 5 (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4966 . . . . 5 ({(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {(card‘𝐴)})
2220, 21syl 17 . . . 4 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ {(card‘𝐴)})
23 fvex 6891 . . . . 5 (card‘𝐴) ∈ V
2423intsn 4983 . . . 4 {(card‘𝐴)} = (card‘𝐴)
2522, 24sseqtrdi 4028 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ (card‘𝐴))
261, 25eqsstrd 4016 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27 cff 10225 . . . . . 6 cf:On⟶On
2827fdmi 6716 . . . . 5 dom cf = On
2928eleq2i 2824 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6913 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3129, 30sylnbir 330 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
32 0ss 4392 . . 3 ∅ ⊆ (card‘𝐴)
3331, 32eqsstrdi 4032 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
3426, 33pm2.61i 182 1 (cf‘𝐴) ⊆ (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2708  wral 3060  wrex 3069  wss 3944  c0 4318  {csn 4622   cint 4943  dom cdm 5669  Oncon0 6353  cfv 6532  cardccrd 9912  cfccf 9914
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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-card 9916  df-cf 9918
This theorem is referenced by:  cfle  10231
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