MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankf Structured version   Visualization version   GIF version

Theorem rankf 9215
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 9186 . . . 4 rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
21funmpt2 6387 . . 3 Fun rank
3 mptv 5162 . . . . . 6 (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
41, 3eqtri 2842 . . . . 5 rank = {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
54dmeqi 5766 . . . 4 dom rank = dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
6 dmopab 5777 . . . . 5 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = {𝑥 ∣ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}}
7 abeq1 2944 . . . . . 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 9214 . . . . . . 7 (𝑥 (𝑅1 “ On) ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
9 intexrab 5234 . . . . . . 7 (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
10 isset 3505 . . . . . . 7 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V ↔ ∃𝑧 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
118, 9, 103bitrri 300 . . . . . 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 2842 . . . 4 dom {⟨𝑥, 𝑧⟩ ∣ 𝑧 = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}} = (𝑅1 “ On)
145, 13eqtri 2842 . . 3 dom rank = (𝑅1 “ On)
15 df-fn 6351 . . 3 (rank Fn (𝑅1 “ On) ↔ (Fun rank ∧ dom rank = (𝑅1 “ On)))
162, 14, 15mpbir2an 709 . 2 rank Fn (𝑅1 “ On)
17 rabn0 4337 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘suc 𝑦))
188, 17bitr4i 280 . . . 4 (𝑥 (𝑅1 “ On) ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅)
19 intex 5231 . . . . . 6 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ ↔ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V)
20 vex 3496 . . . . . . 7 𝑥 ∈ V
211fvmpt2 6772 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V) → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2220, 21mpan 688 . . . . . 6 ( {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ V → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
2319, 22sylbi 219 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
24 ssrab2 4054 . . . . . 6 {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On
25 oninton 7507 . . . . . 6 (({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅) → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2624, 25mpan 688 . . . . 5 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ∈ On)
2723, 26eqeltrd 2911 . . . 4 ({𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)} ≠ ∅ → (rank‘𝑥) ∈ On)
2818, 27sylbi 219 . . 3 (𝑥 (𝑅1 “ On) → (rank‘𝑥) ∈ On)
2928rgen 3146 . 2 𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On
30 ffnfv 6875 . 2 (rank: (𝑅1 “ On)⟶On ↔ (rank Fn (𝑅1 “ On) ∧ ∀𝑥 (𝑅1 “ On)(rank‘𝑥) ∈ On))
3116, 29, 30mpbir2an 709 1 rank: (𝑅1 “ On)⟶On
Colors of variables: wff setvar class
Syntax hints:  wb 208   = wceq 1531  wex 1774  wcel 2108  {cab 2797  wne 3014  wral 3136  wrex 3137  {crab 3140  Vcvv 3493  wss 3934  c0 4289   cuni 4830   cint 4867  {copab 5119  cmpt 5137  dom cdm 5548  cima 5551  Oncon0 6184  suc csuc 6186  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  𝑅1cr1 9183  rankcrnk 9184
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-r1 9185  df-rank 9186
This theorem is referenced by:  rankon  9216  rankvaln  9220  tcrank  9305  hsmexlem4  9843  hsmexlem5  9844  grur1  10234  aomclem4  39647
  Copyright terms: Public domain W3C validator