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Theorem rankwflemb 8607
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 4410 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)))
2 r1funlim 8580 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
32simpli 474 . . . . . . 7 Fun 𝑅1
4 fvelima 6210 . . . . . . 7 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
53, 4mpan 705 . . . . . 6 (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
6 eleq2 2687 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦 → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴𝑦))
76biimprcd 240 . . . . . . . 8 (𝐴𝑦 → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1𝑥)))
8 r1tr 8590 . . . . . . . . . . . 12 Tr (𝑅1𝑥)
9 trss 4726 . . . . . . . . . . . 12 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
108, 9ax-mp 5 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
11 elpwg 4143 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → (𝐴 ∈ 𝒫 (𝑅1𝑥) ↔ 𝐴 ⊆ (𝑅1𝑥)))
1210, 11mpbird 247 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
13 elfvdm 6182 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → 𝑥 ∈ dom 𝑅1)
14 r1sucg 8583 . . . . . . . . . . 11 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1513, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1612, 15eleqtrrd 2701 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥))
1716a1i 11 . . . . . . . 8 (𝑥 ∈ On → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ (𝑅1‘suc 𝑥)))
187, 17syl9 77 . . . . . . 7 (𝐴𝑦 → (𝑥 ∈ On → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1‘suc 𝑥))))
1918reximdvai 3010 . . . . . 6 (𝐴𝑦 → (∃𝑥 ∈ On (𝑅1𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
205, 19syl5 34 . . . . 5 (𝐴𝑦 → (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
2120imp 445 . . . 4 ((𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2221exlimiv 1855 . . 3 (∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
231, 22sylbi 207 . 2 (𝐴 (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
24 elfvdm 6182 . . . . . 6 (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
25 fvelrn 6313 . . . . . 6 ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
263, 24, 25sylancr 694 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
27 df-ima 5092 . . . . . 6 (𝑅1 “ On) = ran (𝑅1 ↾ On)
28 funrel 5869 . . . . . . . . 9 (Fun 𝑅1 → Rel 𝑅1)
293, 28ax-mp 5 . . . . . . . 8 Rel 𝑅1
302simpri 478 . . . . . . . . 9 Lim dom 𝑅1
31 limord 5748 . . . . . . . . 9 (Lim dom 𝑅1 → Ord dom 𝑅1)
32 ordsson 6943 . . . . . . . . 9 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
3330, 31, 32mp2b 10 . . . . . . . 8 dom 𝑅1 ⊆ On
34 relssres 5401 . . . . . . . 8 ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
3529, 33, 34mp2an 707 . . . . . . 7 (𝑅1 ↾ On) = 𝑅1
3635rneqi 5317 . . . . . 6 ran (𝑅1 ↾ On) = ran 𝑅1
3727, 36eqtri 2643 . . . . 5 (𝑅1 “ On) = ran 𝑅1
3826, 37syl6eleqr 2709 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
39 elunii 4412 . . . 4 ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 (𝑅1 “ On))
4038, 39mpdan 701 . . 3 (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4140rexlimivw 3023 . 2 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4223, 41impbii 199 1 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  wss 3559  𝒫 cpw 4135   cuni 4407  Tr wtr 4717  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Rel wrel 5084  Ord word 5686  Oncon0 5687  Lim wlim 5688  suc csuc 5689  Fun wfun 5846  cfv 5852  𝑅1cr1 8576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-r1 8578
This theorem is referenced by:  rankf  8608  r1elwf  8610  rankvalb  8611  rankidb  8614  rankwflem  8629  tcrank  8698  dfac12r  8919
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