Theorem cardf2 8769

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 8765	. . . 4 ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
2	1	funmpt2 5927	. . 3 ⊢ Fun card
3		rabab 3223	. . . 4 ⊢ {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
4	1	dmmpt 5630	. . . 4 ⊢ dom card = {𝑥 ∈ V ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
5		intexrab 4823	. . . . 5 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V)
6	5	abbii 2739	. . . 4 ⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V}
7	3, 4, 6	3eqtr4i 2654	. . 3 ⊢ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
8		df-fn 5891	. . 3 ⊢ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}))
9	2, 7, 8	mpbir2an 955	. 2 ⊢ card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}
10		simpr 477	. . . . . . . . 9 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
11		vex 3203	. . . . . . . . 9 ⊢ 𝑤 ∈ V
12	10, 11	syl6eqelr 2710	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
13		intex 4820	. . . . . . . 8 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V)
14	12, 13	sylibr 224	. . . . . . 7 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅)
15		rabn0 3958	. . . . . . 7 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
16	14, 15	sylib 208	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
17		vex 3203	. . . . . . 7 ⊢ 𝑧 ∈ V
18		breq2 4657	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧))
19	18	rexbidv 3052	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧))
20	17, 19	elab 3350	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)
21	16, 20	sylibr 224	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥})
22		ssrab2 3687	. . . . . . 7 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On
23		oninton 7000	. . . . . . 7 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
24	22, 14, 23	sylancr 695	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On)
25	10, 24	eqeltrd 2701	. . . . 5 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 ∈ On)
26	21, 25	jca 554	. . . 4 ⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On))
27	26	ssopab2i 5003	. . 3 ⊢ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
28		df-card 8765	. . . 4 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
29		df-mpt 4730	. . . 4 ⊢ (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
30	28, 29	eqtri 2644	. . 3 ⊢ card = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})}
31		df-xp 5120	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)}
32	27, 30, 31	3sstr4i 3644	. 2 ⊢ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On)
33		dff2 6371	. 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 955	1 ⊢ card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884   ≈ cen 7952  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-fun 5890  df-fn 5891  df-f 5892  df-card 8765

This theorem is referenced by:  cardon  8770  isnum2  8771  cardf  9372  smobeth  9408  hashkf  13119  hashgval  13120