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Theorem rankf 8914
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 8885 . . . 4 rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
21funmpt2 6150 . . 3 Fun rank
3 mptv 4956 . . . . . 6 (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
41, 3eqtri 2839 . . . . 5 rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
54dmeqi 5540 . . . 4 dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6 dmopab 5550 . . . . 5 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7 abeq1 2928 . . . . . 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 8913 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9 intexrab 5028 . . . . . . 7 (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10 isset 3412 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
118, 9, 103bitrri 289 . . . . . 6 (∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 (𝑅1 “ On))
127, 11mpgbir 1881 . . . . 5 {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
136, 12eqtri 2839 . . . 4 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
145, 13eqtri 2839 . . 3 dom rank = (𝑅1 “ On)
15 df-fn 6114 . . 3 (rank Fn (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = (𝑅1 “ On)))
162, 14, 15mpbir2an 693 . 2 rank Fn (𝑅1 “ On)
17 rabn0 4169 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
188, 17bitr4i 269 . . . 4 (𝑥 (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19 intex 5025 . . . . . 6 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20 vex 3405 . . . . . . 7 𝑥 ∈ V
211fvmpt2 6522 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2220, 21mpan 673 . . . . . 6 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2319, 22sylbi 208 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24 ssrab2 3895 . . . . . 6 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25 oninton 7240 . . . . . 6 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2624, 25mpan 673 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2723, 26eqeltrd 2896 . . . 4 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
2818, 27sylbi 208 . . 3 (𝑥 (𝑅1 “ On) → (rank‘𝑥) ∈ On)
2928rgen 3121 . 2 𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On
30 ffnfv 6620 . 2 (rank: (𝑅1 “ On)⟶On ↔ (rank Fn (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On))
3116, 29, 30mpbir2an 693 1 rank: (𝑅1 “ On)⟶On
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1637  wex 1859  wcel 2157  {cab 2803  wne 2989  wral 3107  wrex 3108  {crab 3111  Vcvv 3402  wss 3780  c0 4127   cuni 4641   cint 4680  {copab 4917  cmpt 4934  dom cdm 5324  cima 5327  Oncon0 5950  suc csuc 5952  Fun wfun 6105   Fn wfn 6106  wf 6107  cfv 6111  𝑅1cr1 8882  rankcrnk 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-om 7306  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-r1 8884  df-rank 8885
This theorem is referenced by:  rankon  8915  rankvaln  8919  tcrank  9004  hsmexlem4  9546  hsmexlem5  9547  grur1  9937  aomclem4  38146
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