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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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