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Theorem rankuni2b 9878
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 9844 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
2 rankval3b 9851 . . . 4 ( 𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31, 2sylbi 216 . . 3 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
4 eleq2 2814 . . . . . 6 (𝑧 = 𝑥𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
54ralbidv 3167 . . . . 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 2810 . . . . . 6 (𝑦 = 𝐴 → ( 𝑥𝑦 (rank‘𝑥) ∈ On ↔ 𝑥𝐴 (rank‘𝑥) ∈ On))
8 vex 3465 . . . . . . 7 𝑦 ∈ V
9 rankon 9820 . . . . . . . 8 (rank‘𝑥) ∈ On
109rgenw 3054 . . . . . . 7 𝑥𝑦 (rank‘𝑥) ∈ On
11 iunon 8360 . . . . . . 7 ((𝑦 ∈ V ∧ ∀𝑥𝑦 (rank‘𝑥) ∈ On) → 𝑥𝑦 (rank‘𝑥) ∈ On)
128, 10, 11mp2an 690 . . . . . 6 𝑥𝑦 (rank‘𝑥) ∈ On
137, 12vtoclg 3532 . . . . 5 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ On)
14 eluni2 4913 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
15 nfv 1909 . . . . . . . 8 𝑥 𝐴 (𝑅1 “ On)
16 nfiu1 5031 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
1716nfel2 2910 . . . . . . . 8 𝑥(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)
18 r1elssi 9830 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
1918sseld 3975 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (𝑥𝐴𝑥 (𝑅1 “ On)))
20 rankelb 9849 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
2119, 20syl6 35 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
22 ssiun2 5051 . . . . . . . . . . 11 (𝑥𝐴 → (rank‘𝑥) ⊆ 𝑥𝐴 (rank‘𝑥))
2322sseld 3975 . . . . . . . . . 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 3253 . . . . . . 7 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2714, 26biimtrid 241 . . . . . 6 (𝐴 (𝑅1 “ On) → (𝑦 𝐴 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2827ralrimiv 3134 . . . . 5 (𝐴 (𝑅1 “ On) → ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))
295, 13, 28elrabd 3681 . . . 4 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
30 intss1 4967 . . . 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 4015 . 2 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) ⊆ 𝑥𝐴 (rank‘𝑥))
331biimpi 215 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
34 elssuni 4941 . . . . 5 (𝑥𝐴𝑥 𝐴)
35 rankssb 9873 . . . . 5 ( 𝐴 (𝑅1 “ On) → (𝑥 𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3633, 34, 35syl2im 40 . . . 4 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3736ralrimiv 3134 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
38 iunss 5049 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
3937, 38sylibr 233 . 2 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4032, 39eqssd 3994 1 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  wss 3944   cuni 4909   cint 4950   ciun 4997  cima 5681  Oncon0 6371  cfv 6549  𝑅1cr1 9787  rankcrnk 9788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-rank 9790
This theorem is referenced by:  rankuni2  9880  rankcf  10802
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