Theorem rankf 9707

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 9678	. . . 4 ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2	1	funmpt2 6526	. . 3 ⊢ Fun rank
3		mptv 5180	. . . . . 6 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
4	1, 3	eqtri 2758	. . . . 5 ⊢ rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
5	4	dmeqi 5848	. . . 4 ⊢ dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6		dmopab 5859	. . . . 5 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7		eqabcb 2875	. . . . . 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 9706	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9		intexrab 5277	. . . . . . 7 ⊢ (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10		isset 3441	. . . . . . 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 1801	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
13	6, 12	eqtri 2758	. . . 4 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
14	5, 13	eqtri 2758	. . 3 ⊢ dom rank = ∪ (𝑅1 “ On)
15		df-fn 6490	. . 3 ⊢ (rank Fn ∪ (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = ∪ (𝑅1 “ On)))
16	2, 14, 15	mpbir2an 712	. 2 ⊢ rank Fn ∪ (𝑅1 “ On)
17		rabn0 4319	. . . . 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 5274	. . . . . 6 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20		vex 3431	. . . . . . 7 ⊢ 𝑥 ∈ V
21	1	fvmpt2 6948	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
22	20, 21	mpan 691	. . . . . 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 4013	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25		oninton 7738	. . . . . 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 691	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
27	23, 26	eqeltrd 2835	. . . 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 3051	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On
30		ffnfv 7060	. 2 ⊢ (rank:∪ (𝑅1 “ On)⟶On ↔ (rank Fn ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On))
31	16, 29, 30	mpbir2an 712	1 ⊢ rank:∪ (𝑅1 “ On)⟶On

Syntax hints:   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2713   ≠ wne 2930  ∀wral 3049  ∃wrex 3059  {crab 3387  Vcvv 3427   ⊆ wss 3885  ∅c0 4263  ∪ cuni 4840  ∩ cint 4879  {copab 5136   ↦ cmpt 5155  dom cdm 5620   “ cima 5623  Oncon0 6312  suc csuc 6314  Fun wfun 6481   Fn wfn 6482  ⟶wf 6483  ‘cfv 6487  𝑅₁cr1 9675  rankcrnk 9676

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-r1 9677  df-rank 9678

This theorem is referenced by:  rankon  9708  rankvaln  9712  tcrank  9797  hsmexlem4  10340  hsmexlem5  10341  grur1  10732  regsfromunir1  36710  aomclem4  43473  rankrelp  45375