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Theorem cardf2 8713
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 8709 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21funmpt2 5885 . . 3 Fun card
3 rabab 3209 . . . 4 {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
41dmmpt 5589 . . . 4 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
5 intexrab 4783 . . . . 5 (∃𝑦 ∈ On 𝑦𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V)
65abbii 2736 . . . 4 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
73, 4, 63eqtr4i 2653 . . 3 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
8 df-fn 5850 . . 3 (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}))
92, 7, 8mpbir2an 954 . 2 card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
10 simpr 477 . . . . . . . . 9 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})
11 vex 3189 . . . . . . . . 9 𝑤 ∈ V
1210, 11syl6eqelr 2707 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
13 intex 4780 . . . . . . . 8 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
1412, 13sylibr 224 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅)
15 rabn0 3932 . . . . . . 7 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦𝑧)
1614, 15sylib 208 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → ∃𝑦 ∈ On 𝑦𝑧)
17 vex 3189 . . . . . . 7 𝑧 ∈ V
18 breq2 4617 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1918rexbidv 3045 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦𝑥 ↔ ∃𝑦 ∈ On 𝑦𝑧))
2017, 19elab 3333 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ ∃𝑦 ∈ On 𝑦𝑧)
2116, 20sylibr 224 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥})
22 ssrab2 3666 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On
23 oninton 6947 . . . . . . 7 (({𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2422, 14, 23sylancr 694 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ On)
2510, 24eqeltrd 2698 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 ∈ On)
2621, 25jca 554 . . . 4 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On))
2726ssopab2i 4963 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
28 df-card 8709 . . . 4 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
29 df-mpt 4675 . . . 4 (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
3028, 29eqtri 2643 . . 3 card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
31 df-xp 5080 . . 3 ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
3227, 30, 313sstr4i 3623 . 2 card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)
33 dff2 6327 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)))
349, 32, 33mpbir2an 954 1 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891   cint 4440   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  dom cdm 5074  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  wf 5843  cen 7896  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-fun 5849  df-fn 5850  df-f 5851  df-card 8709
This theorem is referenced by:  cardon  8714  isnum2  8715  cardf  9316  smobeth  9352  hashkf  13059  hashgval  13060
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