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Theorem rankf 8770
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.
StepHypRef Expression
1 df-rank 8741 . . . 4 rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
21funmpt2 6040 . . 3 Fun rank
3 mptv 4859 . . . . . 6 (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
41, 3eqtri 2746 . . . . 5 rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
54dmeqi 5432 . . . 4 dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6 dmopab 5442 . . . . 5 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7 abeq1 2835 . . . . . 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 8769 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9 intexrab 4928 . . . . . . 7 (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10 isset 3311 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
118, 9, 103bitrri 287 . . . . . 6 (∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 (𝑅1 “ On))
127, 11mpgbir 1839 . . . . 5 {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
136, 12eqtri 2746 . . . 4 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
145, 13eqtri 2746 . . 3 dom rank = (𝑅1 “ On)
15 df-fn 6004 . . 3 (rank Fn (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = (𝑅1 “ On)))
162, 14, 15mpbir2an 993 . 2 rank Fn (𝑅1 “ On)
17 rabn0 4066 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
188, 17bitr4i 267 . . . 4 (𝑥 (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19 intex 4925 . . . . . 6 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20 vex 3307 . . . . . . 7 𝑥 ∈ V
211fvmpt2 6405 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2220, 21mpan 708 . . . . . 6 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2319, 22sylbi 207 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24 ssrab2 3793 . . . . . 6 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25 oninton 7117 . . . . . 6 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2624, 25mpan 708 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2723, 26eqeltrd 2803 . . . 4 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
2818, 27sylbi 207 . . 3 (𝑥 (𝑅1 “ On) → (rank‘𝑥) ∈ On)
2928rgen 3024 . 2 𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On
30 ffnfv 6503 . 2 (rank: (𝑅1 “ On)⟶On ↔ (rank Fn (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On))
3116, 29, 30mpbir2an 993 1 rank: (𝑅1 “ On)⟶On
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1596  wex 1817  wcel 2103  {cab 2710  wne 2896  wral 3014  wrex 3015  {crab 3018  Vcvv 3304  wss 3680  c0 4023   cuni 4544   cint 4583  {copab 4820  cmpt 4837  dom cdm 5218  cima 5221  Oncon0 5836  suc csuc 5838  Fun wfun 5995   Fn wfn 5996  wf 5997  cfv 6001  𝑅1cr1 8738  rankcrnk 8739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-r1 8740  df-rank 8741
This theorem is referenced by:  rankon  8771  rankvaln  8775  tcrank  8860  hsmexlem4  9364  hsmexlem5  9365  grur1  9755  aomclem4  38046
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