Theorem cardf2 9632

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 9628	. . . 4 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
2	1	funmpt2 6457	. . 3 ⊢ Fun card
3		rabab 3450	. . . 4 ⊢ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
4	1	dmmpt 6132	. . . 4 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
5		intexrab 5259	. . . . 5 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V)
6	5	abbii 2809	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
7	3, 4, 6	3eqtr4i 2776	. . 3 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
8		df-fn 6421	. . 3 ⊢ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}))
9	2, 7, 8	mpbir2an 707	. 2 ⊢ card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
10		simpr 484	. . . . . . . . 9 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
11		vex 3426	. . . . . . . . 9 ⊢ 𝑤 ∈ V
12	10, 11	eqeltrrdi 2848	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
13		intex 5256	. . . . . . . 8 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
14	12, 13	sylibr 233	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅)
15		rabn0 4316	. . . . . . 7 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
16	14, 15	sylib 217	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
17		vex 3426	. . . . . . 7 ⊢ 𝑧 ∈ V
18		breq2 5074	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧))
19	18	rexbidv 3225	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧))
20	17, 19	elab 3602	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
21	16, 20	sylibr 233	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥})
22		ssrab2 4009	. . . . . . 7 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On
23		oninton 7622	. . . . . . 7 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
24	22, 14, 23	sylancr 586	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
25	10, 24	eqeltrd 2839	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 ∈ On)
26	21, 25	jca 511	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On))
27	26	ssopab2i 5456	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
28		df-card 9628	. . . 4 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
29		df-mpt 5154	. . . 4 ⊢ (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
30	28, 29	eqtri 2766	. . 3 ⊢ card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
31		df-xp 5586	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
32	27, 30, 31	3sstr4i 3960	. 2 ⊢ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On)
33		dff2 6957	. 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 707	1 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414   ≈ cen 8688  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-fun 6420  df-fn 6421  df-f 6422  df-card 9628

This theorem is referenced by:  cardon  9633  isnum2  9634  cardf  10237  smobeth  10273  hashkf  13974  hashgval  13975  cardpred  32962  nummin  32963