Theorem cardf2 9929

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 9925	. . . 4 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
2	1	funmpt2 6576	. . 3 ⊢ Fun card
3		rabab 3493	. . . 4 ⊢ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
4	1	dmmpt 6242	. . . 4 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
5		intexrab 5318	. . . . 5 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V)
6	5	abbii 2836	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
7	3, 4, 6	3eqtr4i 2802	. . 3 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
8		df-fn 6540	. . 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 3467	. . . . . . . . 9 ⊢ 𝑤 ∈ V
12	10, 11	eqeltrrdi 2878	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
13		intex 5315	. . . . . . . 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 4353	. . . . . . 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 3467	. . . . . . 7 ⊢ 𝑧 ∈ V
18		breq2 5117	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧))
19	18	rexbidv 3195	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧))
20	17, 19	elab 3647	. . . . . 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 4042	. . . . . . 7 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On
23		oninton 7794	. . . . . . 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 2869	. . . . 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 5536	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
28		df-card 9925	. . . 4 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
29		df-mpt 5197	. . . 4 ⊢ (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
30	28, 29	eqtri 2792	. . 3 ⊢ card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
31		df-xp 5668	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
32	27, 30, 31	3sstr4i 3996	. 2 ⊢ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On)
33		dff2 7095	. 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 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ∩ cint 4916   class class class wbr 5113  {copab 5177   ↦ cmpt 5196   × cxp 5660  dom cdm 5662  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533   ≈ cen 8940  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-fun 6539  df-fn 6540  df-f 6541  df-card 9925

This theorem is referenced by:  cardon  9930  isnum2  9931  cardf  10534  smobeth  10571  hashkf  14368  hashgval  14369  cardpred  35426  nummin  35427