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Theorem rankval3b 8853
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 8822 . . . . . . . . . 10 (rank‘𝐴) ∈ On
2 simprl 754 . . . . . . . . . 10 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3 ontri1 5898 . . . . . . . . . 10 (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
41, 2, 3sylancr 575 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
54con2bid 343 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ⊆ 𝑥))
6 r1elssi 8832 . . . . . . . . . . . . . . . . . 18 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
76adantr 466 . . . . . . . . . . . . . . . . 17 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
87sselda 3752 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑦 (𝑅1 “ On))
9 rankdmr1 8828 . . . . . . . . . . . . . . . . . 18 (rank‘𝐴) ∈ dom 𝑅1
10 r1funlim 8793 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 473 . . . . . . . . . . . . . . . . . . 19 Lim dom 𝑅1
12 limord 5925 . . . . . . . . . . . . . . . . . . 19 (Lim dom 𝑅1 → Ord dom 𝑅1)
13 ordtr1 5908 . . . . . . . . . . . . . . . . . . 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 671 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
1615ad2antlr 706 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑥 ∈ dom 𝑅1)
17 rankr1ag 8829 . . . . . . . . . . . . . . . 16 ((𝑦 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
188, 16, 17syl2anc 573 . . . . . . . . . . . . . . 15 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
1918ralbidva 3134 . . . . . . . . . . . . . 14 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → (∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥) ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥))
2019biimpar 463 . . . . . . . . . . . . 13 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2120an32s 631 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
22 dfss3 3741 . . . . . . . . . . . 12 (𝐴 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2321, 22sylibr 224 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1𝑥))
24 simpll 750 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
2515adantl 467 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝑥 ∈ dom 𝑅1)
26 rankr1bg 8830 . . . . . . . . . . . 12 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2724, 25, 26syl2anc 573 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2823, 27mpbid 222 . . . . . . . . . 10 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → (rank‘𝐴) ⊆ 𝑥)
2928ex 397 . . . . . . . . 9 ((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
3029adantrl 695 . . . . . . . 8 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (𝑥 ∈ (rank‘𝐴) → (rank‘𝐴) ⊆ 𝑥))
315, 30sylbird 250 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (¬ (rank‘𝐴) ⊆ 𝑥 → (rank‘𝐴) ⊆ 𝑥))
3231pm2.18d 125 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → (rank‘𝐴) ⊆ 𝑥)
3332ex 397 . . . . 5 (𝐴 (𝑅1 “ On) → ((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3433alrimiv 2007 . . . 4 (𝐴 (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35 ssintab 4628 . . . 4 ((rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)} ↔ ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
3634, 35sylibr 224 . . 3 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)})
37 df-rab 3070 . . . 4 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3837inteqi 4615 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3936, 38syl6sseqr 3801 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
40 rankelb 8851 . . . 4 (𝐴 (𝑅1 “ On) → (𝑦𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
4140ralrimiv 3114 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42 eleq2 2839 . . . . 5 (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
4342ralbidv 3135 . . . 4 (𝑥 = (rank‘𝐴) → (∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
4443onintss 5916 . . 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 3769 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 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  {crab 3065  wss 3723   cuni 4574   cint 4611  dom cdm 5249  cima 5252  Ord word 5863  Oncon0 5864  Lim wlim 5865  Fun wfun 6023  cfv 6029  𝑅1cr1 8789  rankcrnk 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792
This theorem is referenced by:  ranksnb  8854  rankonidlem  8855  rankval3  8867  rankunb  8877  rankuni2b  8880  tcrank  8911
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