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Theorem rankuni2b 8889
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 8855 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
2 rankval3b 8862 . . . 4 ( 𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31, 2sylbi 207 . . 3 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
4 iuneq1 4686 . . . . . . 7 (𝑦 = 𝐴 𝑥𝑦 (rank‘𝑥) = 𝑥𝐴 (rank‘𝑥))
54eleq1d 2824 . . . . . 6 (𝑦 = 𝐴 → ( 𝑥𝑦 (rank‘𝑥) ∈ On ↔ 𝑥𝐴 (rank‘𝑥) ∈ On))
6 vex 3343 . . . . . . 7 𝑦 ∈ V
7 rankon 8831 . . . . . . . 8 (rank‘𝑥) ∈ On
87rgenw 3062 . . . . . . 7 𝑥𝑦 (rank‘𝑥) ∈ On
9 iunon 7605 . . . . . . 7 ((𝑦 ∈ V ∧ ∀𝑥𝑦 (rank‘𝑥) ∈ On) → 𝑥𝑦 (rank‘𝑥) ∈ On)
106, 8, 9mp2an 710 . . . . . 6 𝑥𝑦 (rank‘𝑥) ∈ On
115, 10vtoclg 3406 . . . . 5 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ On)
12 eluni2 4592 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
13 nfv 1992 . . . . . . . 8 𝑥 𝐴 (𝑅1 “ On)
14 nfiu1 4702 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
1514nfel2 2919 . . . . . . . 8 𝑥(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)
16 r1elssi 8841 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
1716sseld 3743 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (𝑥𝐴𝑥 (𝑅1 “ On)))
18 rankelb 8860 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
1917, 18syl6 35 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
20 ssiun2 4715 . . . . . . . . . . 11 (𝑥𝐴 → (rank‘𝑥) ⊆ 𝑥𝐴 (rank‘𝑥))
2120sseld 3743 . . . . . . . . . 10 (𝑥𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2221a1i 11 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → ((rank‘𝑦) ∈ (rank‘𝑥) → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))))
2319, 22syldd 72 . . . . . . . 8 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))))
2413, 15, 23rexlimd 3164 . . . . . . 7 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2512, 24syl5bi 232 . . . . . 6 (𝐴 (𝑅1 “ On) → (𝑦 𝐴 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2625ralrimiv 3103 . . . . 5 (𝐴 (𝑅1 “ On) → ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))
27 eleq2 2828 . . . . . . 7 (𝑧 = 𝑥𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2827ralbidv 3124 . . . . . 6 (𝑧 = 𝑥𝐴 (rank‘𝑥) → (∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2928elrab 3504 . . . . 5 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} ↔ ( 𝑥𝐴 (rank‘𝑥) ∈ On ∧ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
3011, 26, 29sylanbrc 701 . . . 4 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31 intss1 4644 . . . 4 ( 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} → {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ 𝑥𝐴 (rank‘𝑥))
3230, 31syl 17 . . 3 (𝐴 (𝑅1 “ On) → {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧} ⊆ 𝑥𝐴 (rank‘𝑥))
333, 32eqsstrd 3780 . 2 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) ⊆ 𝑥𝐴 (rank‘𝑥))
341biimpi 206 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 elssuni 4619 . . . . 5 (𝑥𝐴𝑥 𝐴)
36 rankssb 8884 . . . . 5 ( 𝐴 (𝑅1 “ On) → (𝑥 𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3734, 35, 36syl2im 40 . . . 4 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3837ralrimiv 3103 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
39 iunss 4713 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4038, 39sylibr 224 . 2 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4133, 40eqssd 3761 1 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715   cuni 4588   cint 4627   ciun 4672  cima 5269  Oncon0 5884  cfv 6049  𝑅1cr1 8798  rankcrnk 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801
This theorem is referenced by:  rankuni2  8891  rankcf  9791
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