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Theorem rankwflemb 8829
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 4591 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)))
2 r1funlim 8802 . . . . . . . 8 (Fun 𝑅1 ∧ Lim dom 𝑅1)
32simpli 476 . . . . . . 7 Fun 𝑅1
4 fvelima 6410 . . . . . . 7 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
53, 4mpan 708 . . . . . 6 (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
6 eleq2 2828 . . . . . . . . 9 ((𝑅1𝑥) = 𝑦 → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴𝑦))
76biimprcd 240 . . . . . . . 8 (𝐴𝑦 → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1𝑥)))
8 r1tr 8812 . . . . . . . . . . . 12 Tr (𝑅1𝑥)
9 trss 4913 . . . . . . . . . . . 12 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
108, 9ax-mp 5 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
11 elpwg 4310 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → (𝐴 ∈ 𝒫 (𝑅1𝑥) ↔ 𝐴 ⊆ (𝑅1𝑥)))
1210, 11mpbird 247 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
13 elfvdm 6381 . . . . . . . . . . 11 (𝐴 ∈ (𝑅1𝑥) → 𝑥 ∈ dom 𝑅1)
14 r1sucg 8805 . . . . . . . . . . 11 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1513, 14syl 17 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
1612, 15eleqtrrd 2842 . . . . . . . . 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 3153 . . . . . 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 444 . . . 4 ((𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2221exlimiv 2007 . . 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 6381 . . . . . 6 (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
25 fvelrn 6515 . . . . . 6 ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
263, 24, 25sylancr 698 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
27 df-ima 5279 . . . . . 6 (𝑅1 “ On) = ran (𝑅1 ↾ On)
28 funrel 6066 . . . . . . . . 9 (Fun 𝑅1 → Rel 𝑅1)
293, 28ax-mp 5 . . . . . . . 8 Rel 𝑅1
302simpri 481 . . . . . . . . 9 Lim dom 𝑅1
31 limord 5945 . . . . . . . . 9 (Lim dom 𝑅1 → Ord dom 𝑅1)
32 ordsson 7154 . . . . . . . . 9 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
3330, 31, 32mp2b 10 . . . . . . . 8 dom 𝑅1 ⊆ On
34 relssres 5595 . . . . . . . 8 ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
3529, 33, 34mp2an 710 . . . . . . 7 (𝑅1 ↾ On) = 𝑅1
3635rneqi 5507 . . . . . 6 ran (𝑅1 ↾ On) = ran 𝑅1
3727, 36eqtri 2782 . . . . 5 (𝑅1 “ On) = ran 𝑅1
3826, 37syl6eleqr 2850 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
39 elunii 4593 . . . 4 ((𝐴 ∈ (𝑅1‘suc 𝑥) ∧ (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On)) → 𝐴 (𝑅1 “ On))
4038, 39mpdan 705 . . 3 (𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4140rexlimivw 3167 . 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 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  wss 3715  𝒫 cpw 4302   cuni 4588  Tr wtr 4904  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Rel wrel 5271  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886  Fun wfun 6043  cfv 6049  𝑅1cr1 8798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800
This theorem is referenced by:  rankf  8830  r1elwf  8832  rankvalb  8833  rankidb  8836  rankwflem  8851  tcrank  8920  dfac12r  9160
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