MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankuni2b Structured version   Visualization version   GIF version

Theorem rankuni2b 8576
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 8542 . . . 4 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
2 rankval3b 8549 . . . 4 ( 𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31, 2sylbi 205 . . 3 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
4 iuneq1 4464 . . . . . . 7 (𝑦 = 𝐴 𝑥𝑦 (rank‘𝑥) = 𝑥𝐴 (rank‘𝑥))
54eleq1d 2671 . . . . . 6 (𝑦 = 𝐴 → ( 𝑥𝑦 (rank‘𝑥) ∈ On ↔ 𝑥𝐴 (rank‘𝑥) ∈ On))
6 vex 3175 . . . . . . 7 𝑦 ∈ V
7 rankon 8518 . . . . . . . 8 (rank‘𝑥) ∈ On
87rgenw 2907 . . . . . . 7 𝑥𝑦 (rank‘𝑥) ∈ On
9 iunon 7300 . . . . . . 7 ((𝑦 ∈ V ∧ ∀𝑥𝑦 (rank‘𝑥) ∈ On) → 𝑥𝑦 (rank‘𝑥) ∈ On)
106, 8, 9mp2an 703 . . . . . 6 𝑥𝑦 (rank‘𝑥) ∈ On
115, 10vtoclg 3238 . . . . 5 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ On)
12 eluni2 4370 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
13 nfv 1829 . . . . . . . 8 𝑥 𝐴 (𝑅1 “ On)
14 nfiu1 4480 . . . . . . . . 9 𝑥 𝑥𝐴 (rank‘𝑥)
1514nfel2 2766 . . . . . . . 8 𝑥(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)
16 r1elssi 8528 . . . . . . . . . . 11 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
1716sseld 3566 . . . . . . . . . 10 (𝐴 (𝑅1 “ On) → (𝑥𝐴𝑥 (𝑅1 “ On)))
18 rankelb 8547 . . . . . . . . . 10 (𝑥 (𝑅1 “ On) → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥)))
1917, 18syl6 34 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ (rank‘𝑥))))
20 ssiun2 4493 . . . . . . . . . . 11 (𝑥𝐴 → (rank‘𝑥) ⊆ 𝑥𝐴 (rank‘𝑥))
2120sseld 3566 . . . . . . . . . 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 69 . . . . . . . 8 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))))
2413, 15, 23rexlimd 3007 . . . . . . 7 (𝐴 (𝑅1 “ On) → (∃𝑥𝐴 𝑦𝑥 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2512, 24syl5bi 230 . . . . . 6 (𝐴 (𝑅1 “ On) → (𝑦 𝐴 → (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2625ralrimiv 2947 . . . . 5 (𝐴 (𝑅1 “ On) → ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥))
27 eleq2 2676 . . . . . . 7 (𝑧 = 𝑥𝐴 (rank‘𝑥) → ((rank‘𝑦) ∈ 𝑧 ↔ (rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2827ralbidv 2968 . . . . . 6 (𝑧 = 𝑥𝐴 (rank‘𝑥) → (∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑥𝐴 (rank‘𝑥)))
2928elrab 3330 . . . . 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 694 . . . 4 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ∈ {𝑧 ∈ On ∣ ∀𝑦 𝐴(rank‘𝑦) ∈ 𝑧})
31 intss1 4421 . . . 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 3601 . 2 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) ⊆ 𝑥𝐴 (rank‘𝑥))
341biimpi 204 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
35 elssuni 4397 . . . . 5 (𝑥𝐴𝑥 𝐴)
36 rankssb 8571 . . . . 5 ( 𝐴 (𝑅1 “ On) → (𝑥 𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3734, 35, 36syl2im 39 . . . 4 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ⊆ (rank‘ 𝐴)))
3837ralrimiv 2947 . . 3 (𝐴 (𝑅1 “ On) → ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
39 iunss 4491 . . 3 ( 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴) ↔ ∀𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4038, 39sylibr 222 . 2 (𝐴 (𝑅1 “ On) → 𝑥𝐴 (rank‘𝑥) ⊆ (rank‘ 𝐴))
4133, 40eqssd 3584 1 (𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  wss 3539   cuni 4366   cint 4404   ciun 4449  cima 5031  Oncon0 5626  cfv 5790  𝑅1cr1 8485  rankcrnk 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-r1 8487  df-rank 8488
This theorem is referenced by:  rankuni2  8578  rankcf  9455
  Copyright terms: Public domain W3C validator