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

Theorem rankwflemb 9836
Description: Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rankwflemb (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankwflemb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 4916 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)))
2 eleq2 2815 . . . . . . . 8 ((𝑅1𝑥) = 𝑦 → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴𝑦))
32biimprcd 249 . . . . . . 7 (𝐴𝑦 → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1𝑥)))
4 r1tr 9819 . . . . . . . . . . 11 Tr (𝑅1𝑥)
5 trss 5281 . . . . . . . . . . 11 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
64, 5ax-mp 5 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
7 elpwg 4610 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → (𝐴 ∈ 𝒫 (𝑅1𝑥) ↔ 𝐴 ⊆ (𝑅1𝑥)))
86, 7mpbird 256 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
9 elfvdm 6938 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝑥 ∈ dom 𝑅1)
10 r1sucg 9812 . . . . . . . . . 10 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
119, 10syl 17 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
128, 11eleqtrrd 2829 . . . . . . . 8 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥))
1312a1i 11 . . . . . . 7 (𝑥 ∈ On → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥)))
143, 13syl9 77 . . . . . 6 (𝐴𝑦 → (𝑥 ∈ On → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1‘suc 𝑥))))
1514reximdvai 3155 . . . . 5 (𝐴𝑦 → (∃𝑥 ∈ On (𝑅1𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16 r1funlim 9809 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1716simpli 482 . . . . . 6 Fun 𝑅1
18 fvelima 6968 . . . . . 6 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
1917, 18mpan 688 . . . . 5 (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
2015, 19impel 504 . . . 4 ((𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2120exlimiv 1926 . . 3 (∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
221, 21sylbi 216 . 2 (𝐴 (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
23 elfvdm 6938 . . . . . 6 (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 fvelrn 7090 . . . . . 6 ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
2517, 23, 24sylancr 585 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26 df-ima 5695 . . . . . 6 (𝑅1 “ On) = ran (𝑅1 ↾ On)
27 funrel 6576 . . . . . . . . 9 (Fun 𝑅1 → Rel 𝑅1)
2817, 27ax-mp 5 . . . . . . . 8 Rel 𝑅1
2916simpri 484 . . . . . . . . 9 Lim dom 𝑅1
30 limord 6436 . . . . . . . . 9 (Lim dom 𝑅1 → Ord dom 𝑅1)
31 ordsson 7791 . . . . . . . . 9 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
3229, 30, 31mp2b 10 . . . . . . . 8 dom 𝑅1 ⊆ On
33 relssres 6031 . . . . . . . 8 ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
3428, 32, 33mp2an 690 . . . . . . 7 (𝑅1 ↾ On) = 𝑅1
3534rneqi 5943 . . . . . 6 ran (𝑅1 ↾ On) = ran 𝑅1
3626, 35eqtri 2754 . . . . 5 (𝑅1 “ On) = ran 𝑅1
3725, 36eleqtrrdi 2837 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
38 elunii 4918 . . . 4 ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 (𝑅1 “ On))
3937, 38mpdan 685 . . 3 (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4039rexlimivw 3141 . 2 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4122, 40impbii 208 1 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wrex 3060  wss 3947  𝒫 cpw 4607   cuni 4913  Tr wtr 5270  dom cdm 5682  ran crn 5683  cres 5684  cima 5685  Rel wrel 5687  Ord word 6375  Oncon0 6376  Lim wlim 6377  suc csuc 6378  Fun wfun 6548  cfv 6554  𝑅1cr1 9805
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 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-r1 9807
This theorem is referenced by:  rankf  9837  r1elwf  9839  rankvalb  9840  rankidb  9843  rankwflem  9858  tcrank  9927  dfac12r  10189
  Copyright terms: Public domain W3C validator