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Theorem rankf 9812
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 9783 . . . 4 rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
21funmpt2 6587 . . 3 Fun rank
3 mptv 5259 . . . . . 6 (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
41, 3eqtri 2756 . . . . 5 rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
54dmeqi 5902 . . . 4 dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6 dmopab 5913 . . . . 5 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7 eqabcb 2871 . . . . . 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 9811 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9 intexrab 5337 . . . . . . 7 (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10 isset 3483 . . . . . . 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 1794 . . . . 5 {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
136, 12eqtri 2756 . . . 4 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
145, 13eqtri 2756 . . 3 dom rank = (𝑅1 “ On)
15 df-fn 6546 . . 3 (rank Fn (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = (𝑅1 “ On)))
162, 14, 15mpbir2an 710 . 2 rank Fn (𝑅1 “ On)
17 rabn0 4382 . . . . 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 5334 . . . . . 6 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20 vex 3474 . . . . . . 7 𝑥 ∈ V
211fvmpt2 7011 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2220, 21mpan 689 . . . . . 6 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2319, 22sylbi 216 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24 ssrab2 4074 . . . . . 6 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25 oninton 7793 . . . . . 6 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2624, 25mpan 689 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2723, 26eqeltrd 2829 . . . 4 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
2818, 27sylbi 216 . . 3 (𝑥 (𝑅1 “ On) → (rank‘𝑥) ∈ On)
2928rgen 3059 . 2 𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On
30 ffnfv 7124 . 2 (rank: (𝑅1 “ On)⟶On ↔ (rank Fn (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On))
3116, 29, 30mpbir2an 710 1 rank: (𝑅1 “ On)⟶On
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wne 2936  wral 3057  wrex 3066  {crab 3428  Vcvv 3470  wss 3945  c0 4319   cuni 4904   cint 4945  {copab 5205  cmpt 5226  dom cdm 5673  cima 5676  Oncon0 6364  suc csuc 6366  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  𝑅1cr1 9780  rankcrnk 9781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-om 7866  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-r1 9782  df-rank 9783
This theorem is referenced by:  rankon  9813  rankvaln  9817  tcrank  9902  hsmexlem4  10447  hsmexlem5  10448  grur1  10838  aomclem4  42472
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