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Theorem rankuni2b 9891
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, 8-Jun-2013.)
Assertion
Ref Expression
rankuni2b (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rankuni2b
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniwf 9857 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
2 rankval3b 9864 . . . 4 ( 𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31, 2sylbi 217 . . 3 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
4 eleq2 2828 . . . . . 6 (𝑧 = 𝑥𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
54ralbidv 3176 . . . . 5 (𝑧 = 𝑥𝐴 (rank‘𝑥) → (∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
6 iuneq1 5013 . . . . . . 7 (𝑦 = 𝐴 𝑥𝑦 (rank‘𝑥) = 𝑥𝐴 (rank‘𝑥))
76eleq1d 2824 . . . . . 6 (𝑦 = 𝐴 → ( 𝑥𝑦 (rank‘𝑥) ∈ On ↔ 𝑥𝐴 (rank‘𝑥) ∈ On))
8 vex 3482 . . . . . . 7 𝑦 ∈ V
9 rankon 9833 . . . . . . . 8 (rank‘𝑥) ∈ On
109rgenw 3063 . . . . . . 7 𝑥𝑦 (rank‘𝑥) ∈ On
11 iunon 8378 . . . . . . 7 ((𝑦 ∈ V ∧ ∀𝑥𝑦 (rank‘𝑥) ∈ On) → 𝑥𝑦 (rank‘𝑥) ∈ On)
128, 10, 11mp2an 692 . . . . . 6 𝑥𝑦 (rank‘𝑥) ∈ On
137, 12vtoclg 3554 . . . . 5 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ On)
14 eluni2 4916 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
15 nfv 1912 . . . . . . . 8 𝑥 𝐴 (𝑅1 “ On)
16 nfiu1 5032 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
1716nfel2 2922 . . . . . . . 8 𝑥(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)
18 r1elssi 9843 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
1918sseld 3994 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (𝑥𝐴𝑥 (𝑅1 “ On)))
20 rankelb 9862 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
2119, 20syl6 35 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
22 ssiun2 5052 . . . . . . . . . . 11 (𝑥𝐴 → (rank‘𝑥) ⊆ 𝑥𝐴 (rank‘𝑥))
2322sseld 3994 . . . . . . . . . 10 (𝑥𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2423a1i 11 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))))
2521, 24syldd 72 . . . . . . . 8 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))))
2615, 17, 25rexlimd 3264 . . . . . . 7 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2714, 26biimtrid 242 . . . . . 6 (𝐴 (𝑅1 “ On) → (𝑦 𝐴 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2827ralrimiv 3143 . . . . 5 (𝐴 (𝑅1 “ On) → ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))
295, 13, 28elrabd 3697 . . . 4 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
30 intss1 4968 . . . 4 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} → {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ 𝑥𝐴 (rank‘𝑥))
3129, 30syl 17 . . 3 (𝐴 (𝑅1 “ On) → {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ 𝑥𝐴 (rank‘𝑥))
323, 31eqsstrd 4034 . 2 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) ⊆ 𝑥𝐴 (rank‘𝑥))
331biimpi 216 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
34 elssuni 4942 . . . . 5 (𝑥𝐴𝑥 𝐴)
35 rankssb 9886 . . . . 5 ( 𝐴 (𝑅1 “ On) → (𝑥 𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3633, 34, 35syl2im 40 . . . 4 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3736ralrimiv 3143 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
38 iunss 5050 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
3937, 38sylibr 234 . 2 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4032, 39eqssd 4013 1 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963   cuni 4912   cint 4951   ciun 4996  cima 5692  Oncon0 6386  cfv 6563  𝑅1cr1 9800  rankcrnk 9801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803
This theorem is referenced by:  rankuni2  9893  rankcf  10815
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