Theorem rankf 9754

Description: The domain and codomain 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 9725	. . . 4 ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2	1	funmpt2 6558	. . 3 ⊢ Fun rank
3		mptv 5216	. . . . . 6 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
4	1, 3	eqtri 2753	. . . . 5 ⊢ rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
5	4	dmeqi 5871	. . . 4 ⊢ dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6		dmopab 5882	. . . . 5 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7		eqabcb 2870	. . . . . 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 9753	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9		intexrab 5305	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10		isset 3464	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
11	8, 9, 10	3bitrri 298	. . . . . 6 ⊢ (∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
12	7, 11	mpgbir 1799	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
13	6, 12	eqtri 2753	. . . 4 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
14	5, 13	eqtri 2753	. . 3 ⊢ dom rank = ∪ (𝑅1 “ On)
15		df-fn 6517	. . 3 ⊢ (rank Fn ∪ (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = ∪ (𝑅1 “ On)))
16	2, 14, 15	mpbir2an 711	. 2 ⊢ rank Fn ∪ (𝑅1 “ On)
17		rabn0 4355	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
18	8, 17	bitr4i 278	. . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19		intex 5302	. . . . . 6 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
21	1	fvmpt2 6982	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
22	20, 21	mpan 690	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
23	19, 22	sylbi 217	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24		ssrab2 4046	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25		oninton 7774	. . . . . 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 690	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
27	23, 26	eqeltrd 2829	. . . 4 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
28	18, 27	sylbi 217	. . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) ∈ On)
29	28	rgen 3047	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On
30		ffnfv 7094	. 2 ⊢ (rank:∪ (𝑅1 “ On)⟶On ↔ (rank Fn ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On))
31	16, 29, 30	mpbir2an 711	1 ⊢ rank:∪ (𝑅1 “ On)⟶On

Syntax hints:   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ∪ cuni 4874  ∩ cint 4913  {copab 5172   ↦ cmpt 5191  dom cdm 5641   “ cima 5644  Oncon0 6335  suc csuc 6337  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  𝑅₁cr1 9722  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-r1 9724  df-rank 9725

This theorem is referenced by:  rankon  9755  rankvaln  9759  tcrank  9844  hsmexlem4  10389  hsmexlem5  10390  grur1  10780  aomclem4  43053  rankrelp  44957