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Theorem rankval3b 8852
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 8821 . . . . . . . . . 10 (rank‘𝐴) ∈ On
2 simprl 746 . . . . . . . . . 10 ((𝐴 (𝑅1 “ On) ∧ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)) → 𝑥 ∈ On)
3 ontri1 5900 . . . . . . . . . 10 (((rank‘𝐴) ∈ On ∧ 𝑥 ∈ On) → ((rank‘𝐴) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ (rank‘𝐴)))
41, 2, 3sylancr 567 . . . . . . . . 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 8831 . . . . . . . . . . . . . . . . . 18 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
76adantr 466 . . . . . . . . . . . . . . . . 17 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
87sselda 3750 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑦 (𝑅1 “ On))
9 rankdmr1 8827 . . . . . . . . . . . . . . . . . 18 (rank‘𝐴) ∈ dom 𝑅1
10 r1funlim 8792 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑅1 ∧ Lim dom 𝑅1)
1110simpri 473 . . . . . . . . . . . . . . . . . . 19 Lim dom 𝑅1
12 limord 5927 . . . . . . . . . . . . . . . . . . 19 (Lim dom 𝑅1 → Ord dom 𝑅1)
13 ordtr1 5910 . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (rank‘𝐴) → 𝑥 ∈ dom 𝑅1)
1615ad2antlr 698 . . . . . . . . . . . . . . . 16 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → 𝑥 ∈ dom 𝑅1)
17 rankr1ag 8828 . . . . . . . . . . . . . . . 16 ((𝑦 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
188, 16, 17syl2anc 565 . . . . . . . . . . . . . . 15 (((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ (rank‘𝐴)) ∧ 𝑦𝐴) → (𝑦 ∈ (𝑅1𝑥) ↔ (rank‘𝑦) ∈ 𝑥))
1918ralbidva 3133 . . . . . . . . . . . . . 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 623 . . . . . . . . . . . 12 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
22 dfss3 3739 . . . . . . . . . . . 12 (𝐴 ⊆ (𝑅1𝑥) ↔ ∀𝑦𝐴 𝑦 ∈ (𝑅1𝑥))
2321, 22sylibr 224 . . . . . . . . . . 11 (((𝐴 (𝑅1 “ On) ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) ∧ 𝑥 ∈ (rank‘𝐴)) → 𝐴 ⊆ (𝑅1𝑥))
24 simpll 742 . . . . . . . . . . . 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 8829 . . . . . . . . . . . 12 ((𝐴 (𝑅1 “ On) ∧ 𝑥 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝑥) ↔ (rank‘𝐴) ⊆ 𝑥))
2724, 25, 26syl2anc 565 . . . . . . . . . . 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 687 . . . . . . . 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 2006 . . . 4 (𝐴 (𝑅1 “ On) → ∀𝑥((𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥) → (rank‘𝐴) ⊆ 𝑥))
35 ssintab 4626 . . . 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 3069 . . . 4 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3837inteqi 4613 . . 3 {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥} = {𝑥 ∣ (𝑥 ∈ On ∧ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥)}
3936, 38syl6sseqr 3799 . 2 (𝐴 (𝑅1 “ On) → (rank‘𝐴) ⊆ {𝑥 ∈ On ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})
40 rankelb 8850 . . . 4 (𝐴 (𝑅1 “ On) → (𝑦𝐴 → (rank‘𝑦) ∈ (rank‘𝐴)))
4140ralrimiv 3113 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴))
42 eleq2 2838 . . . . 5 (𝑥 = (rank‘𝐴) → ((rank‘𝑦) ∈ 𝑥 ↔ (rank‘𝑦) ∈ (rank‘𝐴)))
4342ralbidv 3134 . . . 4 (𝑥 = (rank‘𝐴) → (∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥 ↔ ∀𝑦𝐴 (rank‘𝑦) ∈ (rank‘𝐴)))
4443onintss 5918 . . 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 3767 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 1628   = wceq 1630  wcel 2144  {cab 2756  wral 3060  {crab 3064  wss 3721   cuni 4572   cint 4609  dom cdm 5249  cima 5252  Ord word 5865  Oncon0 5866  Lim wlim 5867  Fun wfun 6025  cfv 6031  𝑅1cr1 8788  rankcrnk 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-r1 8790  df-rank 8791
This theorem is referenced by:  ranksnb  8853  rankonidlem  8854  rankval3  8866  rankunb  8876  rankuni2b  8879  tcrank  8910
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