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Theorem rankval3b 9866
Description: The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankval3b (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem rankval3b
StepHypRef Expression
1 rankon 9835 . . . . . . . . . 10 (rank‘𝐴) ∈ On
2 simprl 771 . . . . . . . . . 10 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3 ontri1 6418 . . . . . . . . . 10 (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
41, 2, 3sylancr 587 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
54con2bid 354 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6 r1elssi 9845 . . . . . . . . . . . . . . . . . 18 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
76adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
87sselda 3983 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑦 (𝑅1 “ On))
9 rankdmr1 9841 . . . . . . . . . . . . . . . . . 18 (rank‘𝐴) ∈ dom 𝑅1
10 r1funlim 9806 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 485 . . . . . . . . . . . . . . . . . . 19 Lim dom 𝑅1
12 limord 6444 . . . . . . . . . . . . . . . . . . 19 (Lim dom 𝑅1 → Ord dom 𝑅1)
13 ordtr1 6427 . . . . . . . . . . . . . . . . . . 19 (Ord dom 𝑅1 → ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1))
1411, 12, 13mp2b 10 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ (rank‘𝐴) ∧ (rank‘𝐴) ∈ dom 𝑅1) → 𝑥 ∈ dom 𝑅1)
159, 14mpan2 691 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
1615ad2antlr 727 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑥 ∈ dom 𝑅1)
17 rankr1ag 9842 . . . . . . . . . . . . . . . 16 ((𝑦 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
188, 16, 17syl2anc 584 . . . . . . . . . . . . . . 15 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
1918ralbidva 3176 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥) ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥))
2019biimpar 477 . . . . . . . . . . . . 13 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2120an32s 652 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
22 dfss3 3972 . . . . . . . . . . . 12 (𝐴 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2321, 22sylibr 234 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1𝑥))
24 simpll 767 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
2515adantl 481 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26 rankr1bg 9843 . . . . . . . . . . . 12 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2724, 25, 26syl2anc 584 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2823, 27mpbid 232 . . . . . . . . . 10 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
2928ex 412 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
3029adantrl 716 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
315, 30sylbird 260 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
3231pm2.18d 127 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
3332ex 412 . . . . 5 (𝐴 (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3433alrimiv 1927 . . . 4 (𝐴 (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35 ssintab 4965 . . . 4 ((rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3634, 35sylibr 234 . . 3 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)})
37 df-rab 3437 . . . 4 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3837inteqi 4950 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3936, 38sseqtrrdi 4025 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
40 rankelb 9864 . . . 4 (𝐴 (𝑅1 “ On) → (𝑦𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
4140ralrimiv 3145 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42 eleq2 2830 . . . . 5 (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
4342ralbidv 3178 . . . 4 (𝑥 = (rank‘𝐴) → (∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
4443onintss 6435 . . 3 ((rank‘𝐴) ∈ On → (∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴) → {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴)))
451, 41, 44mpsyl 68 . 2 (𝐴 (𝑅1 “ On) → {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} ⊆ (rank‘𝐴))
4639, 45eqssd 4001 1 (𝐴 (𝑅1 “ On) → (rank‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  wss 3951   cuni 4907   cint 4946  dom cdm 5685  cima 5688  Ord word 6383  Oncon0 6384  Lim wlim 6385  Fun wfun 6555  cfv 6561  𝑅1cr1 9802  rankcrnk 9803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805
This theorem is referenced by:  ranksnb  9867  rankonidlem  9868  rankval3  9880  rankunb  9890  rankuni2b  9893  tcrank  9924
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