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Theorem rankuni2b 9117
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 9083 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
2 rankval3b 9090 . . . 4 ( 𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31, 2sylbi 218 . . 3 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
4 eleq2 2869 . . . . . 6 (𝑧 = 𝑥𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
54ralbidv 3162 . . . . 5 (𝑧 = 𝑥𝐴 (rank‘𝑥) → (∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
6 iuneq1 4834 . . . . . . 7 (𝑦 = 𝐴 𝑥𝑦 (rank‘𝑥) = 𝑥𝐴 (rank‘𝑥))
76eleq1d 2865 . . . . . 6 (𝑦 = 𝐴 → ( 𝑥𝑦 (rank‘𝑥) ∈ On ↔ 𝑥𝐴 (rank‘𝑥) ∈ On))
8 vex 3435 . . . . . . 7 𝑦 ∈ V
9 rankon 9059 . . . . . . . 8 (rank‘𝑥) ∈ On
109rgenw 3115 . . . . . . 7 𝑥𝑦 (rank‘𝑥) ∈ On
11 iunon 7819 . . . . . . 7 ((𝑦 ∈ V ∧ ∀𝑥𝑦 (rank‘𝑥) ∈ On) → 𝑥𝑦 (rank‘𝑥) ∈ On)
128, 10, 11mp2an 688 . . . . . 6 𝑥𝑦 (rank‘𝑥) ∈ On
137, 12vtoclg 3505 . . . . 5 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ On)
14 eluni2 4743 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
15 nfv 1890 . . . . . . . 8 𝑥 𝐴 (𝑅1 “ On)
16 nfiu1 4850 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
1716nfel2 2963 . . . . . . . 8 𝑥(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)
18 r1elssi 9069 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
1918sseld 3883 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (𝑥𝐴𝑥 (𝑅1 “ On)))
20 rankelb 9088 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
2119, 20syl6 35 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
22 ssiun2 4864 . . . . . . . . . . 11 (𝑥𝐴 → (rank‘𝑥) ⊆ 𝑥𝐴 (rank‘𝑥))
2322sseld 3883 . . . . . . . . . 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 3275 . . . . . . 7 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2714, 26syl5bi 243 . . . . . 6 (𝐴 (𝑅1 “ On) → (𝑦 𝐴 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2827ralrimiv 3146 . . . . 5 (𝐴 (𝑅1 “ On) → ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))
295, 13, 28elrabd 3615 . . . 4 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
30 intss1 4791 . . . 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 3921 . 2 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) ⊆ 𝑥𝐴 (rank‘𝑥))
331biimpi 217 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
34 elssuni 4768 . . . . 5 (𝑥𝐴𝑥 𝐴)
35 rankssb 9112 . . . . 5 ( 𝐴 (𝑅1 “ On) → (𝑥 𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3633, 34, 35syl2im 40 . . . 4 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3736ralrimiv 3146 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
38 iunss 4862 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
3937, 38sylibr 235 . 2 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4032, 39eqssd 3901 1 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079  wral 3103  wrex 3104  {crab 3107  Vcvv 3432  wss 3854   cuni 4739   cint 4776   ciun 4819  cima 5438  Oncon0 6058  cfv 6217  𝑅1cr1 9026  rankcrnk 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-om 7428  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-r1 9028  df-rank 9029
This theorem is referenced by:  rankuni2  9119  rankcf  10034
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