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Theorem cardf2 8652
 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 8648 . . . 4 card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
21funmpt2 5841 . . 3 Fun card
3 rabab 3196 . . . 4 {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
41dmmpt 5547 . . . 4 dom card = {𝑥 ∈ V ∣ {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
5 intexrab 4750 . . . . 5 (∃𝑦 ∈ On 𝑦𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V)
65abbii 2726 . . . 4 {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} = {𝑥 {𝑦 ∈ On ∣ 𝑦𝑥} ∈ V}
73, 4, 63eqtr4i 2642 . . 3 dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
8 df-fn 5807 . . 3 (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}))
92, 7, 8mpbir2an 957 . 2 card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}
10 simpr 476 . . . . . . . . 9 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})
11 vex 3176 . . . . . . . . 9 𝑤 ∈ V
1210, 11syl6eqelr 2697 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
13 intex 4747 . . . . . . . 8 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑦𝑧} ∈ V)
1412, 13sylibr 223 . . . . . . 7 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → {𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅)
15 rabn0 3912 . . . . . . 7 ({𝑦 ∈ On ∣ 𝑦𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦𝑧)
1614, 15sylib 207 . . . . . 6 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → ∃𝑦 ∈ On 𝑦𝑧)
17 vex 3176 . . . . . . 7 𝑧 ∈ V
18 breq2 4587 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦𝑥𝑦𝑧))
1918rexbidv 3034 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦𝑥 ↔ ∃𝑦 ∈ On 𝑦𝑧))
2017, 19elab 3319 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ↔ ∃𝑦 ∈ On 𝑦𝑧)
2116, 20sylibr 223 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥})
22 ssrab2 3650 . . . . . . 7 {𝑦 ∈ On ∣ 𝑦𝑧} ⊆ On
23 oninton 6892 . . . . . . 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 2688 . . . . 5 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → 𝑤 ∈ On)
2621, 25jca 553 . . . 4 ((𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On))
2726ssopab2i 4928 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
28 df-card 8648 . . . 4 card = (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧})
29 df-mpt 4645 . . . 4 (𝑧 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
3028, 29eqtri 2632 . . 3 card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = {𝑦 ∈ On ∣ 𝑦𝑧})}
31 df-xp 5044 . . 3 ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ 𝑤 ∈ On)}
3227, 30, 313sstr4i 3607 . 2 card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)
33 dff2 6279 . 2 (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥} × On)))
349, 32, 33mpbir2an 957 1 card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800   ≈ cen 7838  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-fun 5806  df-fn 5807  df-f 5808  df-card 8648 This theorem is referenced by:  cardon  8653  isnum2  8654  cardf  9251  smobeth  9287  hashkf  12981  hashgval  12982
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