MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardf2 Structured version   Visualization version   GIF version

Theorem cardf2 9984
Description: The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
cardf2 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
Distinct variable group:   𝑥,𝑦

Proof of Theorem cardf2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-card 9980 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21funmpt2 6604 . . 3 Fun card
3 rabab 3511 . . . 4 {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
41dmmpt 6259 . . . 4 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
5 intexrab 5346 . . . . 5 (∃𝑦 ∈ On 𝑦𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V)
65abbii 2808 . . . 4 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
73, 4, 63eqtr4i 2774 . . 3 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
8 df-fn 6563 . . 3 (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}))
92, 7, 8mpbir2an 711 . 2 card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
10 simpr 484 . . . . . . . . 9 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})
11 vex 3483 . . . . . . . . 9 𝑤 ∈ V
1210, 11eqeltrrdi 2849 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
13 intex 5343 . . . . . . . 8 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
1412, 13sylibr 234 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅)
15 rabn0 4388 . . . . . . 7 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦𝑧)
1614, 15sylib 218 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → ∃𝑦 ∈ On 𝑦𝑧)
17 vex 3483 . . . . . . 7 𝑧 ∈ V
18 breq2 5146 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1918rexbidv 3178 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦𝑥 ↔ ∃𝑦 ∈ On 𝑦𝑧))
2017, 19elab 3678 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ ∃𝑦 ∈ On 𝑦𝑧)
2116, 20sylibr 234 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥})
22 ssrab2 4079 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On
23 oninton 7816 . . . . . . 7 (({𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2422, 14, 23sylancr 587 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2510, 24eqeltrd 2840 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 ∈ On)
2621, 25jca 511 . . . 4 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On))
2726ssopab2i 5554 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
28 df-card 9980 . . . 4 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
29 df-mpt 5225 . . . 4 (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
3028, 29eqtri 2764 . . 3 card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
31 df-xp 5690 . . 3 ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
3227, 30, 313sstr4i 4034 . 2 card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)
33 dff2 7118 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)))
349, 32, 33mpbir2an 711 1 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  c0 4332   cint 4945   class class class wbr 5142  {copab 5204  cmpt 5224   × cxp 5682  dom cdm 5684  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  wf 6556  cen 8983  cardccrd 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-fun 6562  df-fn 6563  df-f 6564  df-card 9980
This theorem is referenced by:  cardon  9985  isnum2  9986  cardf  10591  smobeth  10627  hashkf  14372  hashgval  14373  cardpred  35105  nummin  35106
  Copyright terms: Public domain W3C validator