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

Theorem cardf2 9938
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:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On
Distinct variable group:   π‘₯,𝑦

Proof of Theorem cardf2
Dummy variables 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-card 9934 . . . 4 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
21funmpt2 6588 . . 3 Fun card
3 rabab 3503 . . . 4 {π‘₯ ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V} = {π‘₯ ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V}
41dmmpt 6240 . . . 4 dom card = {π‘₯ ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V}
5 intexrab 5341 . . . . 5 (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V)
65abbii 2803 . . . 4 {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} = {π‘₯ ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V}
73, 4, 63eqtr4i 2771 . . 3 dom card = {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}
8 df-fn 6547 . . 3 (card Fn {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ↔ (Fun card ∧ dom card = {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}))
92, 7, 8mpbir2an 710 . 2 card Fn {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}
10 simpr 486 . . . . . . . . 9 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})
11 vex 3479 . . . . . . . . 9 𝑀 ∈ V
1210, 11eqeltrrdi 2843 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} ∈ V)
13 intex 5338 . . . . . . . 8 ({𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} β‰  βˆ… ↔ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} ∈ V)
1412, 13sylibr 233 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} β‰  βˆ…)
15 rabn0 4386 . . . . . . 7 ({𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} β‰  βˆ… ↔ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧)
1614, 15sylib 217 . . . . . 6 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧)
17 vex 3479 . . . . . . 7 𝑧 ∈ V
18 breq2 5153 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (𝑦 β‰ˆ π‘₯ ↔ 𝑦 β‰ˆ 𝑧))
1918rexbidv 3179 . . . . . . 7 (π‘₯ = 𝑧 β†’ (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯ ↔ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧))
2017, 19elab 3669 . . . . . 6 (𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ↔ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧)
2116, 20sylibr 233 . . . . 5 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ 𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯})
22 ssrab2 4078 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} βŠ† On
23 oninton 7783 . . . . . . 7 (({𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} βŠ† On ∧ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} β‰  βˆ…) β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} ∈ On)
2422, 14, 23sylancr 588 . . . . . 6 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} ∈ On)
2510, 24eqeltrd 2834 . . . . 5 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ 𝑀 ∈ On)
2621, 25jca 513 . . . 4 ((𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) β†’ (𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ∧ 𝑀 ∈ On))
2726ssopab2i 5551 . . 3 {βŸ¨π‘§, π‘€βŸ© ∣ (𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})} βŠ† {βŸ¨π‘§, π‘€βŸ© ∣ (𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ∧ 𝑀 ∈ On)}
28 df-card 9934 . . . 4 card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})
29 df-mpt 5233 . . . 4 (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧}) = {βŸ¨π‘§, π‘€βŸ© ∣ (𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})}
3028, 29eqtri 2761 . . 3 card = {βŸ¨π‘§, π‘€βŸ© ∣ (𝑧 ∈ V ∧ 𝑀 = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})}
31 df-xp 5683 . . 3 ({π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} Γ— On) = {βŸ¨π‘§, π‘€βŸ© ∣ (𝑧 ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ∧ 𝑀 ∈ On)}
3227, 30, 313sstr4i 4026 . 2 card βŠ† ({π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} Γ— On)
33 dff2 7101 . 2 (card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On ↔ (card Fn {π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} ∧ card βŠ† ({π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} Γ— On)))
349, 32, 33mpbir2an 710 1 card:{π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯}⟢On
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βˆ© cint 4951   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   Γ— cxp 5675  dom cdm 5677  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540   β‰ˆ cen 8936  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-fun 6546  df-fn 6547  df-f 6548  df-card 9934
This theorem is referenced by:  cardon  9939  isnum2  9940  cardf  10545  smobeth  10581  hashkf  14292  hashgval  14293  cardpred  34093  nummin  34094
  Copyright terms: Public domain W3C validator