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Theorem rankwflemb 9210
 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 4806 . . 3 (𝐴 (𝑅1 “ On) ↔ ∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)))
2 eleq2 2881 . . . . . . . 8 ((𝑅1𝑥) = 𝑦 → (𝐴 ∈ (𝑅1𝑥) ↔ 𝐴𝑦))
32biimprcd 253 . . . . . . 7 (𝐴𝑦 → ((𝑅1𝑥) = 𝑦𝐴 ∈ (𝑅1𝑥)))
4 r1tr 9193 . . . . . . . . . . 11 Tr (𝑅1𝑥)
5 trss 5148 . . . . . . . . . . 11 (Tr (𝑅1𝑥) → (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥)))
64, 5ax-mp 5 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ⊆ (𝑅1𝑥))
7 elpwg 4503 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → (𝐴 ∈ 𝒫 (𝑅1𝑥) ↔ 𝐴 ⊆ (𝑅1𝑥)))
86, 7mpbird 260 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → 𝐴 ∈ 𝒫 (𝑅1𝑥))
9 elfvdm 6681 . . . . . . . . . 10 (𝐴 ∈ (𝑅1𝑥) → 𝑥 ∈ dom 𝑅1)
10 r1sucg 9186 . . . . . . . . . 10 (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
119, 10syl 17 . . . . . . . . 9 (𝐴 ∈ (𝑅1𝑥) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1𝑥))
128, 11eleqtrrd 2896 . . . . . . . 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 3234 . . . . 5 (𝐴𝑦 → (∃𝑥 ∈ On (𝑅1𝑥) = 𝑦 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)))
16 r1funlim 9183 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1716simpli 487 . . . . . 6 Fun 𝑅1
18 fvelima 6710 . . . . . 6 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
1917, 18mpan 689 . . . . 5 (𝑦 ∈ (𝑅1 “ On) → ∃𝑥 ∈ On (𝑅1𝑥) = 𝑦)
2015, 19impel 509 . . . 4 ((𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
2120exlimiv 1931 . . 3 (∃𝑦(𝐴𝑦𝑦 ∈ (𝑅1 “ On)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
221, 21sylbi 220 . 2 (𝐴 (𝑅1 “ On) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
23 elfvdm 6681 . . . . . 6 (𝐴 ∈ (𝑅1‘suc 𝑥) → suc 𝑥 ∈ dom 𝑅1)
24 fvelrn 6825 . . . . . 6 ((Fun 𝑅1 ∧ suc 𝑥 ∈ dom 𝑅1) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
2517, 23, 24sylancr 590 . . . . 5 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ ran 𝑅1)
26 df-ima 5536 . . . . . 6 (𝑅1 “ On) = ran (𝑅1 ↾ On)
27 funrel 6345 . . . . . . . . 9 (Fun 𝑅1 → Rel 𝑅1)
2817, 27ax-mp 5 . . . . . . . 8 Rel 𝑅1
2916simpri 489 . . . . . . . . 9 Lim dom 𝑅1
30 limord 6222 . . . . . . . . 9 (Lim dom 𝑅1 → Ord dom 𝑅1)
31 ordsson 7488 . . . . . . . . 9 (Ord dom 𝑅1 → dom 𝑅1 ⊆ On)
3229, 30, 31mp2b 10 . . . . . . . 8 dom 𝑅1 ⊆ On
33 relssres 5863 . . . . . . . 8 ((Rel 𝑅1 ∧ dom 𝑅1 ⊆ On) → (𝑅1 ↾ On) = 𝑅1)
3428, 32, 33mp2an 691 . . . . . . 7 (𝑅1 ↾ On) = 𝑅1
3534rneqi 5775 . . . . . 6 ran (𝑅1 ↾ On) = ran 𝑅1
3626, 35eqtri 2824 . . . . 5 (𝑅1 “ On) = ran 𝑅1
3725, 36eleqtrrdi 2904 . . . 4 (𝐴 ∈ (𝑅1‘suc 𝑥) → (𝑅1‘suc 𝑥) ∈ (𝑅1 “ On))
38 elunii 4808 . . . 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 3244 . 2 (∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥) → 𝐴 (𝑅1 “ On))
4122, 40impbii 212 1 (𝐴 (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∃wrex 3110   ⊆ wss 3884  𝒫 cpw 4500  ∪ cuni 4803  Tr wtr 5139  dom cdm 5523  ran crn 5524   ↾ cres 5525   “ cima 5526  Rel wrel 5528  Ord word 6162  Oncon0 6163  Lim wlim 6164  suc csuc 6165  Fun wfun 6322  ‘cfv 6328  𝑅1cr1 9179 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-r1 9181 This theorem is referenced by:  rankf  9211  r1elwf  9213  rankvalb  9214  rankidb  9217  rankwflem  9232  tcrank  9301  dfac12r  9561
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