Theorem rankf 9207

Description: The domain and range of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)

Assertion

Ref	Expression
rankf	⊢ rank:∪ (𝑅1 “ On)⟶On

Proof of Theorem rankf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rank 9178	. . . 4 ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2	1	funmpt2 6363	. . 3 ⊢ Fun rank
3		mptv 5135	. . . . . 6 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
4	1, 3	eqtri 2821	. . . . 5 ⊢ rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
5	4	dmeqi 5737	. . . 4 ⊢ dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6		dmopab 5748	. . . . 5 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7		abeq1 2923	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On) ↔ ∀𝑥(∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 ∈ ∪ (𝑅1 “ On)))
8		rankwflemb 9206	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9		intexrab 5207	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10		isset 3453	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
11	8, 9, 10	3bitrri 301	. . . . . 6 ⊢ (∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
12	7, 11	mpgbir 1801	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
13	6, 12	eqtri 2821	. . . 4 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
14	5, 13	eqtri 2821	. . 3 ⊢ dom rank = ∪ (𝑅1 “ On)
15		df-fn 6327	. . 3 ⊢ (rank Fn ∪ (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = ∪ (𝑅1 “ On)))
16	2, 14, 15	mpbir2an 710	. 2 ⊢ rank Fn ∪ (𝑅1 “ On)
17		rabn0 4293	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
18	8, 17	bitr4i 281	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19		intex 5204	. . . . . 6 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20		vex 3444	. . . . . . 7 ⊢ 𝑥 ∈ V
21	1	fvmpt2 6756	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
22	20, 21	mpan 689	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
23	19, 22	sylbi 220	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24		ssrab2 4007	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25		oninton 7495	. . . . . 6 ⊢ (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
26	24, 25	mpan 689	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
27	23, 26	eqeltrd 2890	. . . 4 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
28	18, 27	sylbi 220	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) ∈ On)
29	28	rgen 3116	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On
30		ffnfv 6859	. 2 ⊢ (rank:∪ (𝑅1 “ On)⟶On ↔ (rank Fn ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On))
31	16, 29, 30	mpbir2an 710	1 ⊢ rank:∪ (𝑅1 “ On)⟶On

Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800  ∩ cint 4838  {copab 5092   ↦ cmpt 5110  dom cdm 5519   “ cima 5522  Oncon0 6159  suc csuc 6161  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  𝑅₁cr1 9175  rankcrnk 9176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178

This theorem is referenced by:  rankon  9208  rankvaln  9212  tcrank  9297  hsmexlem4  9840  hsmexlem5  9841  grur1  10231  aomclem4  40001