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Theorem rankval3b 9244
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 9213 . . . . . . . . . 10 (rank‘𝐴) ∈ On
2 simprl 767 . . . . . . . . . 10 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3 ontri1 6219 . . . . . . . . . 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 356 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6 r1elssi 9223 . . . . . . . . . . . . . . . . . 18 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
76adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
87sselda 3966 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑦 (𝑅1 “ On))
9 rankdmr1 9219 . . . . . . . . . . . . . . . . . 18 (rank‘𝐴) ∈ dom 𝑅1
10 r1funlim 9184 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 486 . . . . . . . . . . . . . . . . . . 19 Lim dom 𝑅1
12 limord 6244 . . . . . . . . . . . . . . . . . . 19 (Lim dom 𝑅1 → Ord dom 𝑅1)
13 ordtr1 6228 . . . . . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
1615ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑥 ∈ dom 𝑅1)
17 rankr1ag 9220 . . . . . . . . . . . . . . . 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 3196 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥) ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥))
2019biimpar 478 . . . . . . . . . . . . 13 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2120an32s 648 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
22 dfss3 3955 . . . . . . . . . . . 12 (𝐴 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2321, 22sylibr 235 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1𝑥))
24 simpll 763 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
2515adantl 482 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26 rankr1bg 9221 . . . . . . . . . . . 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 233 . . . . . . . . . 10 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
2928ex 413 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
3029adantrl 712 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
315, 30sylbird 261 . . . . . . 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 413 . . . . 5 (𝐴 (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3433alrimiv 1919 . . . 4 (𝐴 (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35 ssintab 4886 . . . 4 ((rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3634, 35sylibr 235 . . 3 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)})
37 df-rab 3147 . . . 4 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3837inteqi 4873 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3936, 38sseqtrrdi 4017 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
40 rankelb 9242 . . . 4 (𝐴 (𝑅1 “ On) → (𝑦𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
4140ralrimiv 3181 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42 eleq2 2901 . . . . 5 (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
4342ralbidv 3197 . . . 4 (𝑥 = (rank‘𝐴) → (∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
4443onintss 6235 . . 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 3983 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 207  wa 396  wal 1526   = wceq 1528  wcel 2105  {cab 2799  wral 3138  {crab 3142  wss 3935   cuni 4832   cint 4869  dom cdm 5549  cima 5552  Ord word 6184  Oncon0 6185  Lim wlim 6186  Fun wfun 6343  cfv 6349  𝑅1cr1 9180  rankcrnk 9181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-r1 9182  df-rank 9183
This theorem is referenced by:  ranksnb  9245  rankonidlem  9246  rankval3  9258  rankunb  9268  rankuni2b  9271  tcrank  9302
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