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Theorem cardf2 9366
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 9362 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21funmpt2 6389 . . 3 Fun card
3 rabab 3524 . . . 4 {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
41dmmpt 6089 . . . 4 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
5 intexrab 5236 . . . . 5 (∃𝑦 ∈ On 𝑦𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V)
65abbii 2886 . . . 4 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
73, 4, 63eqtr4i 2854 . . 3 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
8 df-fn 6353 . . 3 (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}))
92, 7, 8mpbir2an 709 . 2 card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
10 simpr 487 . . . . . . . . 9 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})
11 vex 3498 . . . . . . . . 9 𝑤 ∈ V
1210, 11eqeltrrdi 2922 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
13 intex 5233 . . . . . . . 8 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
1412, 13sylibr 236 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅)
15 rabn0 4339 . . . . . . 7 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦𝑧)
1614, 15sylib 220 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → ∃𝑦 ∈ On 𝑦𝑧)
17 vex 3498 . . . . . . 7 𝑧 ∈ V
18 breq2 5063 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1918rexbidv 3297 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦𝑥 ↔ ∃𝑦 ∈ On 𝑦𝑧))
2017, 19elab 3667 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ ∃𝑦 ∈ On 𝑦𝑧)
2116, 20sylibr 236 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥})
22 ssrab2 4056 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On
23 oninton 7509 . . . . . . 7 (({𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2422, 14, 23sylancr 589 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2510, 24eqeltrd 2913 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 ∈ On)
2621, 25jca 514 . . . 4 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On))
2726ssopab2i 5430 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
28 df-card 9362 . . . 4 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
29 df-mpt 5140 . . . 4 (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
3028, 29eqtri 2844 . . 3 card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
31 df-xp 5556 . . 3 ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
3227, 30, 313sstr4i 4010 . 2 card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)
33 dff2 6860 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)))
349, 32, 33mpbir2an 709 1 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wcel 2110  {cab 2799  wne 3016  wrex 3139  {crab 3142  Vcvv 3495  wss 3936  c0 4291   cint 4869   class class class wbr 5059  {copab 5121  cmpt 5139   × cxp 5548  dom cdm 5550  Oncon0 6186  Fun wfun 6344   Fn wfn 6345  wf 6346  cen 8500  cardccrd 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-ord 6189  df-on 6190  df-fun 6352  df-fn 6353  df-f 6354  df-card 9362
This theorem is referenced by:  cardon  9367  isnum2  9368  cardf  9966  smobeth  10002  hashkf  13686  hashgval  13687
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