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Theorem rankval3b 8932
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 8901 . . . . . . . . . 10 (rank‘𝐴) ∈ On
2 simprl 778 . . . . . . . . . 10 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3 ontri1 5970 . . . . . . . . . 10 (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
41, 2, 3sylancr 577 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
54con2bid 345 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6 r1elssi 8911 . . . . . . . . . . . . . . . . . 18 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
76adantr 468 . . . . . . . . . . . . . . . . 17 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
87sselda 3798 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑦 (𝑅1 “ On))
9 rankdmr1 8907 . . . . . . . . . . . . . . . . . 18 (rank‘𝐴) ∈ dom 𝑅1
10 r1funlim 8872 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 475 . . . . . . . . . . . . . . . . . . 19 Lim dom 𝑅1
12 limord 5996 . . . . . . . . . . . . . . . . . . 19 (Lim dom 𝑅1 → Ord dom 𝑅1)
13 ordtr1 5979 . . . . . . . . . . . . . . . . . . 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 674 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
1615ad2antlr 709 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑥 ∈ dom 𝑅1)
17 rankr1ag 8908 . . . . . . . . . . . . . . . 16 ((𝑦 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
188, 16, 17syl2anc 575 . . . . . . . . . . . . . . 15 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
1918ralbidva 3173 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥) ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥))
2019biimpar 465 . . . . . . . . . . . . 13 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2120an32s 634 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
22 dfss3 3787 . . . . . . . . . . . 12 (𝐴 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2321, 22sylibr 225 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1𝑥))
24 simpll 774 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
2515adantl 469 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26 rankr1bg 8909 . . . . . . . . . . . 12 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2724, 25, 26syl2anc 575 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2823, 27mpbid 223 . . . . . . . . . 10 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
2928ex 399 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
3029adantrl 698 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
315, 30sylbird 251 . . . . . . 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 399 . . . . 5 (𝐴 (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3433alrimiv 2018 . . . 4 (𝐴 (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35 ssintab 4686 . . . 4 ((rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3634, 35sylibr 225 . . 3 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)})
37 df-rab 3105 . . . 4 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3837inteqi 4673 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3936, 38syl6sseqr 3849 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
40 rankelb 8930 . . . 4 (𝐴 (𝑅1 “ On) → (𝑦𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
4140ralrimiv 3153 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42 eleq2 2874 . . . . 5 (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
4342ralbidv 3174 . . . 4 (𝑥 = (rank‘𝐴) → (∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
4443onintss 5987 . . 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 3815 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 197  wa 384  wal 1635   = wceq 1637  wcel 2156  {cab 2792  wral 3096  {crab 3100  wss 3769   cuni 4630   cint 4669  dom cdm 5311  cima 5314  Ord word 5935  Oncon0 5936  Lim wlim 5937  Fun wfun 6091  cfv 6097  𝑅1cr1 8868  rankcrnk 8869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-om 7292  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-r1 8870  df-rank 8871
This theorem is referenced by:  ranksnb  8933  rankonidlem  8934  rankval3  8946  rankunb  8956  rankuni2b  8959  tcrank  8990
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