Theorem rankf 9802

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 9769	. . . 4 ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2	1	funmpt2 6579	. . 3 ⊢ Fun rank
3		mptv 5210	. . . . . 6 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
4	1, 3	eqtri 2783	. . . . 5 ⊢ rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
5	4	dmeqi 5885	. . . 4 ⊢ dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6		dmopab 5896	. . . . 5 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7		eqabcb 2900	. . . . . 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 9800	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9		intexrab 5307	. . . . . . 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 301	. . . . . 6 ⊢ (∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
12	7, 11	mpgbir 1831	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
13	6, 12	eqtri 2783	. . . 4 ⊢ dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = ∪ (𝑅1 “ On)
14	5, 13	eqtri 2783	. . 3 ⊢ dom rank = ∪ (𝑅1 “ On)
15		df-fn 6541	. . 3 ⊢ (rank Fn ∪ (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = ∪ (𝑅1 “ On)))
16	2, 14, 15	mpbir2an 724	. 2 ⊢ rank Fn ∪ (𝑅1 “ On)
17		rabn0 4338	. . . . 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 5304	. . . . . 6 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
21	1	fvmpt2 7005	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
22	20, 21	mpan 703	. . . . . 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 4005	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25		oninton 7809	. . . . . 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 703	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
27	23, 26	eqeltrd 2860	. . . 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 3078	. 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On
30		ffnfv 7119	. 2 ⊢ (rank:∪ (𝑅1 “ On)⟶On ↔ (rank Fn ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(rank‘𝑥) ∈ On))
31	16, 29, 30	mpbir2an 724	1 ⊢ rank:∪ (𝑅1 “ On)⟶On

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ∪ cuni 4866  ∩ cint 4906  {copab 5166   ↦ cmpt 5185  dom cdm 5650   “ cima 5653  Oncon0 6362  suc csuc 6364  Fun wfun 6532   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  𝑅₁cr1 9766  rankcrnk 9767

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by:  rankon  9803  rankvaln  9807  tcrank  9901  hsmexlem4  10507  hsmexlem5  10508  grur1  10905  rankfo  35735  regsfromunir1  37328  aomclem4  44059  rankrelp  45950