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Theorem rankf 9832
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.
StepHypRef Expression
1 df-rank 9803 . . . 4 rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
21funmpt2 6607 . . 3 Fun rank
3 mptv 5264 . . . . . 6 (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
41, 3eqtri 2763 . . . . 5 rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
54dmeqi 5918 . . . 4 dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6 dmopab 5929 . . . . 5 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7 eqabcb 2881 . . . . . 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 9831 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9 intexrab 5353 . . . . . . 7 (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10 isset 3492 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
118, 9, 103bitrri 298 . . . . . 6 (∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ↔ 𝑥 (𝑅1 “ On))
127, 11mpgbir 1796 . . . . 5 {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
136, 12eqtri 2763 . . . 4 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
145, 13eqtri 2763 . . 3 dom rank = (𝑅1 “ On)
15 df-fn 6566 . . 3 (rank Fn (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = (𝑅1 “ On)))
162, 14, 15mpbir2an 711 . 2 rank Fn (𝑅1 “ On)
17 rabn0 4395 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
188, 17bitr4i 278 . . . 4 (𝑥 (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19 intex 5350 . . . . . 6 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20 vex 3482 . . . . . . 7 𝑥 ∈ V
211fvmpt2 7027 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2220, 21mpan 690 . . . . . 6 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2319, 22sylbi 217 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24 ssrab2 4090 . . . . . 6 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25 oninton 7815 . . . . . 6 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2624, 25mpan 690 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2723, 26eqeltrd 2839 . . . 4 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
2818, 27sylbi 217 . . 3 (𝑥 (𝑅1 “ On) → (rank‘𝑥) ∈ On)
2928rgen 3061 . 2 𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On
30 ffnfv 7139 . 2 (rank: (𝑅1 “ On)⟶On ↔ (rank Fn (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On))
3116, 29, 30mpbir2an 711 1 rank: (𝑅1 “ On)⟶On
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   cuni 4912   cint 4951  {copab 5210  cmpt 5231  dom cdm 5689  cima 5692  Oncon0 6386  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  𝑅1cr1 9800  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by:  rankon  9833  rankvaln  9837  tcrank  9922  hsmexlem4  10467  hsmexlem5  10468  grur1  10858  aomclem4  43046
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