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Theorem cflecard 9156
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 9150 . . 3 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
2 df-sn 4254 . . . . . 6 {(card‘𝐴)} = {𝑥𝑥 = (card‘𝐴)}
3 ssid 3698 . . . . . . . . 9 𝐴𝐴
4 ssid 3698 . . . . . . . . . . 11 𝑧𝑧
5 sseq2 3701 . . . . . . . . . . . 12 (𝑤 = 𝑧 → (𝑧𝑤𝑧𝑧))
65rspcev 3381 . . . . . . . . . . 11 ((𝑧𝐴𝑧𝑧) → ∃𝑤𝐴 𝑧𝑤)
74, 6mpan2 709 . . . . . . . . . 10 (𝑧𝐴 → ∃𝑤𝐴 𝑧𝑤)
87rgen 2992 . . . . . . . . 9 𝑧𝐴𝑤𝐴 𝑧𝑤
93, 8pm3.2i 470 . . . . . . . 8 (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)
10 fveq2 6272 . . . . . . . . . . 11 (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴))
1110eqeq2d 2702 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴)))
12 sseq1 3700 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦𝐴𝐴𝐴))
13 rexeq 3210 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑤𝑦 𝑧𝑤 ↔ ∃𝑤𝐴 𝑧𝑤))
1413ralbidv 3056 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∀𝑧𝐴𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))
1512, 14anbi12d 749 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤) ↔ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)))
1611, 15anbi12d 749 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤))))
1716spcegv 3366 . . . . . . . 8 (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴𝐴 ∧ ∀𝑧𝐴𝑤𝐴 𝑧𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
189, 17mpan2i 715 . . . . . . 7 (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
1918ss2abdv 3749 . . . . . 6 (𝐴 ∈ On → {𝑥𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
202, 19syl5eqss 3723 . . . . 5 (𝐴 ∈ On → {(card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
21 intss 4574 . . . . 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 6282 . . . . 5 (card‘𝐴) ∈ V
2423intsn 4589 . . . 4 {(card‘𝐴)} = (card‘𝐴)
2522, 24syl6sseq 3725 . . 3 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ⊆ (card‘𝐴))
261, 25eqsstrd 3713 . 2 (𝐴 ∈ On → (cf‘𝐴) ⊆ (card‘𝐴))
27 cff 9151 . . . . . 6 cf:On⟶On
2827fdmi 6133 . . . . 5 dom cf = On
2928eleq2i 2763 . . . 4 (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
30 ndmfv 6299 . . . 4 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
3129, 30sylnbir 320 . . 3 𝐴 ∈ On → (cf‘𝐴) = ∅)
32 0ss 4048 . . 3 ∅ ⊆ (card‘𝐴)
3331, 32syl6eqss 3729 . 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 383   = wceq 1564  wex 1785  wcel 2071  {cab 2678  wral 2982  wrex 2983  wss 3648  c0 3991  {csn 4253   cint 4551  dom cdm 5186  Oncon0 5804  cfv 5969  cardccrd 8842  cfccf 8844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-pss 3664  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-tp 4258  df-op 4260  df-uni 4513  df-int 4552  df-br 4729  df-opab 4789  df-mpt 4806  df-tr 4829  df-id 5096  df-eprel 5101  df-po 5107  df-so 5108  df-fr 5145  df-we 5147  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-ord 5807  df-on 5808  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-fv 5977  df-card 8846  df-cf 8848
This theorem is referenced by:  cfle  9157
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