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

Theorem cflecard 9276
 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 9270 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 df-sn 4315 . . . . . 6 {(card‘𝐴)} = {𝑥𝑥 = (card‘𝐴)}
3 ssid 3771 . . . . . . . . 9 𝐴𝐴
4 ssid 3771 . . . . . . . . . . 11 𝑧𝑧
5 sseq2 3774 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
65rspcev 3458 . . . . . . . . . . 11 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
74, 6mpan2 663 . . . . . . . . . 10 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
87rgen 3070 . . . . . . . . 9 𝑧𝐴𝑤𝐴 𝑧𝑤
93, 8pm3.2i 447 . . . . . . . 8 (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
10 fveq2 6332 . . . . . . . . . . 11 (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
1110eqeq2d 2780 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12 sseq1 3773 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
13 rexeq 3287 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
1413ralbidv 3134 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
1512, 14anbi12d 608 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1611, 15anbi12d 608 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))))
1716spcegv 3443 . . . . . . . 8 (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
189, 17mpan2i 669 . . . . . . 7 (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918ss2abdv 3822 . . . . . 6 (𝐴 ∈ On → {𝑥𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
202, 19syl5eqss 3796 . . . . 5 (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4630 . . . . 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 6342 . . . . 5 (card‘𝐴) ∈ V
2423intsn 4645 . . . 4 {(card‘𝐴)} = (card‘𝐴)
2522, 24syl6sseq 3798 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ (card‘𝐴))
261, 25eqsstrd 3786 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27 cff 9271 . . . . . 6 cf:On⟶On
2827fdmi 6192 . . . . 5 dom cf = On
2928eleq2i 2841 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6359 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3129, 30sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
32 0ss 4114 . . 3 ∅ ⊆ (card‘𝐴)
3331, 32syl6eqss 3802 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
3426, 33pm2.61i 176 1 (cf‘𝐴) ⊆ (card‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 382   = wceq 1630  ∃wex 1851   ∈ wcel 2144  {cab 2756  ∀wral 3060  ∃wrex 3061   ⊆ wss 3721  ∅c0 4061  {csn 4314  ∩ cint 4609  dom cdm 5249  Oncon0 5866  ‘cfv 6031  cardccrd 8960  cfccf 8962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-card 8964  df-cf 8966 This theorem is referenced by:  cfle  9277
 Copyright terms: Public domain W3C validator