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Theorem cardf2 9884
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 9880 . . . 4 card = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
21funmpt2 6541 . . 3 Fun card
3 rabab 3472 . . . 4 {π‘₯ ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V} = {π‘₯ ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V}
41dmmpt 6193 . . . 4 dom card = {π‘₯ ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯} ∈ V}
5 intexrab 5298 . . . . 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 6500 . . 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 3448 . . . . . . . . 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 5295 . . . . . . . 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 4346 . . . . . . 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 3448 . . . . . . 7 𝑧 ∈ V
18 breq2 5110 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (𝑦 β‰ˆ π‘₯ ↔ 𝑦 β‰ˆ 𝑧))
1918rexbidv 3172 . . . . . . 7 (π‘₯ = 𝑧 β†’ (βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯ ↔ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝑧))
2017, 19elab 3631 . . . . . 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 4038 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧} βŠ† On
23 oninton 7731 . . . . . . 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 5508 . . 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 9880 . . . 4 card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝑧})
29 df-mpt 5190 . . . 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 5640 . . 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 3988 . 2 card βŠ† ({π‘₯ ∣ βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ π‘₯} Γ— On)
33 dff2 7050 . 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 2940  βˆƒwrex 3070  {crab 3406  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  βˆ© cint 4908   class class class wbr 5106  {copab 5168   ↦ cmpt 5189   Γ— cxp 5632  dom cdm 5634  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493   β‰ˆ cen 8883  cardccrd 9876
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-fun 6499  df-fn 6500  df-f 6501  df-card 9880
This theorem is referenced by:  cardon  9885  isnum2  9886  cardf  10491  smobeth  10527  hashkf  14238  hashgval  14239  cardpred  33751  nummin  33752
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