Theorem rankf 9483

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 9454	. . . 4 ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2	1	funmpt2 6457	. . 3 ⊢ Fun rank
3		mptv 5186	. . . . . 6 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
4	1, 3	eqtri 2766	. . . . 5 ⊢ rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
5	4	dmeqi 5802	. . . 4 ⊢ dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6		dmopab 5813	. . . . 5 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7		abeq1 2872	. . . . . 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 9482	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9		intexrab 5259	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10		isset 3435	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
11	8, 9, 10	3bitrri 297	. . . . . 6 ⊢ (∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
12	7, 11	mpgbir 1803	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
13	6, 12	eqtri 2766	. . . 4 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
14	5, 13	eqtri 2766	. . 3 ⊢ dom rank = ∪ (𝑅1 “ On)
15		df-fn 6421	. . 3 ⊢ (rank Fn ∪ (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = ∪ (𝑅1 “ On)))
16	2, 14, 15	mpbir2an 707	. 2 ⊢ rank Fn ∪ (𝑅1 “ On)
17		rabn0 4316	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
18	8, 17	bitr4i 277	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19		intex 5256	. . . . . 6 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
21	1	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
22	20, 21	mpan 686	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
23	19, 22	sylbi 216	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24		ssrab2 4009	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25		oninton 7622	. . . . . 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 686	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
27	23, 26	eqeltrd 2839	. . . 4 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
28	18, 27	sylbi 216	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) ∈ On)
29	28	rgen 3073	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On
30		ffnfv 6974	. 2 ⊢ (rank:∪ (𝑅1 “ On)⟶On ↔ (rank Fn ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On))
31	16, 29, 30	mpbir2an 707	1 ⊢ rank:∪ (𝑅1 “ On)⟶On

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∩ cint 4876  {copab 5132   ↦ cmpt 5153  dom cdm 5580   “ cima 5583  Oncon0 6251  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  𝑅₁cr1 9451  rankcrnk 9452

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-pow 5283  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-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  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-iun 4923  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-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-r1 9453  df-rank 9454

This theorem is referenced by:  rankon  9484  rankvaln  9488  tcrank  9573  hsmexlem4  10116  hsmexlem5  10117  grur1  10507  aomclem4  40798