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Theorem rankwflemb 9788
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 4912 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)))
2 eleq2 2823 . . . . . . . 8 ((𝑅1𝑥) = 𝑦 → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴𝑦))
32biimprcd 249 . . . . . . 7 (𝐴𝑦 → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1𝑥)))
4 r1tr 9771 . . . . . . . . . . 11 Tr (𝑅1𝑥)
5 trss 5277 . . . . . . . . . . 11 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
64, 5ax-mp 5 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
7 elpwg 4606 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → (𝐴 ∈ 𝒫 (𝑅1𝑥) ↔ 𝐴 ⊆ (𝑅1𝑥)))
86, 7mpbird 257 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
9 elfvdm 6929 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝑥 ∈ dom 𝑅1)
10 r1sucg 9764 . . . . . . . . . 10 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
119, 10syl 17 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
128, 11eleqtrrd 2837 . . . . . . . 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 3166 . . . . 5 (𝐴𝑦 → (∃𝑥 ∈ On (𝑅1𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16 r1funlim 9761 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1716simpli 485 . . . . . 6 Fun 𝑅1
18 fvelima 6958 . . . . . 6 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
1917, 18mpan 689 . . . . 5 (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
2015, 19impel 507 . . . 4 ((𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2120exlimiv 1934 . . 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 6929 . . . . . 6 (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 fvelrn 7079 . . . . . 6 ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
2517, 23, 24sylancr 588 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26 df-ima 5690 . . . . . 6 (𝑅1 “ On) = ran (𝑅1 ↾ On)
27 funrel 6566 . . . . . . . . 9 (Fun 𝑅1 → Rel 𝑅1)
2817, 27ax-mp 5 . . . . . . . 8 Rel 𝑅1
2916simpri 487 . . . . . . . . 9 Lim dom 𝑅1
30 limord 6425 . . . . . . . . 9 (Lim dom 𝑅1 → Ord dom 𝑅1)
31 ordsson 7770 . . . . . . . . 9 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
3229, 30, 31mp2b 10 . . . . . . . 8 dom 𝑅1 ⊆ On
33 relssres 6023 . . . . . . . 8 ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
3428, 32, 33mp2an 691 . . . . . . 7 (𝑅1 ↾ On) = 𝑅1
3534rneqi 5937 . . . . . 6 ran (𝑅1 ↾ On) = ran 𝑅1
3626, 35eqtri 2761 . . . . 5 (𝑅1 “ On) = ran 𝑅1
3725, 36eleqtrrdi 2845 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
38 elunii 4914 . . . 4 ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 (𝑅1 “ On))
3937, 38mpdan 686 . . 3 (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4039rexlimivw 3152 . 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 397   = wceq 1542  wex 1782  wcel 2107  wrex 3071  wss 3949  𝒫 cpw 4603   cuni 4909  Tr wtr 5266  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  Rel wrel 5682  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  Fun wfun 6538  cfv 6544  𝑅1cr1 9757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759
This theorem is referenced by:  rankf  9789  r1elwf  9791  rankvalb  9792  rankidb  9795  rankwflem  9810  tcrank  9879  dfac12r  10141
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