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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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