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Theorem cflecard 9669
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 9663 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 df-sn 4565 . . . . . 6 {(card‘𝐴)} = {𝑥𝑥 = (card‘𝐴)}
3 ssid 3993 . . . . . . . . 9 𝐴𝐴
4 ssid 3993 . . . . . . . . . . 11 𝑧𝑧
5 sseq2 3997 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
65rspcev 3627 . . . . . . . . . . 11 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
74, 6mpan2 687 . . . . . . . . . 10 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
87rgen 3153 . . . . . . . . 9 𝑧𝐴𝑤𝐴 𝑧𝑤
93, 8pm3.2i 471 . . . . . . . 8 (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
10 fveq2 6669 . . . . . . . . . . 11 (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
1110eqeq2d 2837 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12 sseq1 3996 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
13 rexeq 3412 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
1413ralbidv 3202 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
1512, 14anbi12d 630 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1611, 15anbi12d 630 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))))
1716spcegv 3602 . . . . . . . 8 (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
189, 17mpan2i 693 . . . . . . 7 (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918ss2abdv 4048 . . . . . 6 (𝐴 ∈ On → {𝑥𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
202, 19eqsstrid 4019 . . . . 5 (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4895 . . . . 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 6682 . . . . 5 (card‘𝐴) ∈ V
2423intsn 4910 . . . 4 {(card‘𝐴)} = (card‘𝐴)
2522, 24sseqtrdi 4021 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ (card‘𝐴))
261, 25eqsstrd 4009 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27 cff 9664 . . . . . 6 cf:On⟶On
2827fdmi 6523 . . . . 5 dom cf = On
2928eleq2i 2909 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6699 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3129, 30sylnbir 332 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
32 0ss 4354 . . 3 ∅ ⊆ (card‘𝐴)
3331, 32eqsstrdi 4025 . 2 𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
3426, 33pm2.61i 183 1 (cf‘𝐴) ⊆ (card‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wral 3143  wrex 3144  wss 3940  c0 4295  {csn 4564   cint 4874  dom cdm 5554  Oncon0 6190  cfv 6354  cardccrd 9358  cfccf 9360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-card 9362  df-cf 9364
This theorem is referenced by:  cfle  9670
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