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Theorem cflecard 10293
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 10287 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 df-sn 4627 . . . . . 6 {(card‘𝐴)} = {𝑥𝑥 = (card‘𝐴)}
3 ssid 4006 . . . . . . . . 9 𝐴𝐴
4 ssid 4006 . . . . . . . . . . 11 𝑧𝑧
5 sseq2 4010 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
65rspcev 3622 . . . . . . . . . . 11 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
74, 6mpan2 691 . . . . . . . . . 10 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
87rgen 3063 . . . . . . . . 9 𝑧𝐴𝑤𝐴 𝑧𝑤
93, 8pm3.2i 470 . . . . . . . 8 (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
10 fveq2 6906 . . . . . . . . . . 11 (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
1110eqeq2d 2748 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12 sseq1 4009 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
13 rexeq 3322 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
1413ralbidv 3178 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
1512, 14anbi12d 632 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1611, 15anbi12d 632 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))))
1716spcegv 3597 . . . . . . . 8 (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
189, 17mpan2i 697 . . . . . . 7 (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918ss2abdv 4066 . . . . . 6 (𝐴 ∈ On → {𝑥𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
202, 19eqsstrid 4022 . . . . 5 (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4969 . . . . 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 6919 . . . . 5 (card‘𝐴) ∈ V
2423intsn 4984 . . . 4 {(card‘𝐴)} = (card‘𝐴)
2522, 24sseqtrdi 4024 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ (card‘𝐴))
261, 25eqsstrd 4018 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27 cff 10288 . . . . . 6 cf:On⟶On
2827fdmi 6747 . . . . 5 dom cf = On
2928eleq2i 2833 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6941 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3129, 30sylnbir 331 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
32 0ss 4400 . . 3 ∅ ⊆ (card‘𝐴)
3331, 32eqsstrdi 4028 . 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 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951  c0 4333  {csn 4626   cint 4946  dom cdm 5685  Oncon0 6384  cfv 6561  cardccrd 9975  cfccf 9977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-card 9979  df-cf 9981
This theorem is referenced by:  cfle  10294
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