Theorem cardf2 9925

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 𝑦 ≈ 𝑥}⟶On

Distinct variable group:   𝑥,𝑦

Proof of Theorem cardf2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-card 9921	. . . 4 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
2	1	funmpt2 6575	. . 3 ⊢ Fun card
3		rabab 3485	. . . 4 ⊢ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
4	1	dmmpt 6241	. . . 4 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
5		intexrab 5317	. . . . 5 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V)
6	5	abbii 2830	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
7	3, 4, 6	3eqtr4i 2796	. . 3 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
8		df-fn 6539	. . 3 ⊢ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}))
9	2, 7, 8	mpbir2an 723	. 2 ⊢ card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
10		simpr 489	. . . . . . . . 9 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
11		vex 3459	. . . . . . . . 9 ⊢ 𝑤 ∈ V
12	10, 11	eqeltrrdi 2872	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
13		intex 5314	. . . . . . . 8 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
14	12, 13	sylibr 237	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅)
15		rabn0 4346	. . . . . . 7 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
16	14, 15	sylib 221	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
17		vex 3459	. . . . . . 7 ⊢ 𝑧 ∈ V
18		breq2 5113	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧))
19	18	rexbidv 3189	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧))
20	17, 19	elab 3638	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
21	16, 20	sylibr 237	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥})
22		ssrab2 4034	. . . . . . 7 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On
23		oninton 7790	. . . . . . 7 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
24	22, 14, 23	sylancr 598	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
25	10, 24	eqeltrd 2863	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 ∈ On)
26	21, 25	jca 520	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On))
27	26	ssopab2i 5535	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
28		df-card 9921	. . . 4 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
29		df-mpt 5193	. . . 4 ⊢ (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
30	28, 29	eqtri 2786	. . 3 ⊢ card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
31		df-xp 5667	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
32	27, 30, 31	3sstr4i 3988	. 2 ⊢ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On)
33		dff2 7094	. 2 ⊢ (card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On)))
34	9, 32, 33	mpbir2an 723	1 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∩ cint 4912   class class class wbr 5109  {copab 5173   ↦ cmpt 5192   × cxp 5659  dom cdm 5661  Oncon0 6360  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532   ≈ cen 8936  cardccrd 9917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-fun 6538  df-fn 6539  df-f 6540  df-card 9921

This theorem is referenced by:  cardon  9926  isnum2  9927  cardf  10529  smobeth  10566  hashkf  14364  hashgval  14365  cardpred  35483  nummin  35484