Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcriota GIF version

Theorem bdcriota 10333
Description: A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
Hypotheses
Ref Expression
bdcriota.bd BOUNDED 𝜑
bdcriota.ex ∃!𝑥𝑦 𝜑
Assertion
Ref Expression
bdcriota BOUNDED (𝑥𝑦 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bdcriota
Dummy variables 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdcriota.bd . . . . . . . . 9 BOUNDED 𝜑
21ax-bdsb 10272 . . . . . . . 8 BOUNDED [𝑧 / 𝑥]𝜑
3 ax-bdel 10271 . . . . . . . 8 BOUNDED 𝑡𝑧
42, 3ax-bdim 10264 . . . . . . 7 BOUNDED ([𝑧 / 𝑥]𝜑𝑡𝑧)
54ax-bdal 10268 . . . . . 6 BOUNDED𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧)
6 df-ral 2326 . . . . . . . . 9 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
7 impexp 254 . . . . . . . . . . 11 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ (𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
87bicomi 127 . . . . . . . . . 10 ((𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
98albii 1373 . . . . . . . . 9 (∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
106, 9bitri 177 . . . . . . . 8 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
11 sban 1843 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ ([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑))
12 clelsb3 2156 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑥𝑦𝑧𝑦)
1312anbi1i 439 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1411, 13bitri 177 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1514bicomi 127 . . . . . . . . . 10 ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
1615imbi1i 231 . . . . . . . . 9 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1716albii 1373 . . . . . . . 8 (∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1810, 17bitri 177 . . . . . . 7 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
19 df-clab 2041 . . . . . . . . . 10 (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
2019bicomi 127 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)})
2120imbi1i 231 . . . . . . . 8 (([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2221albii 1373 . . . . . . 7 (∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2318, 22bitri 177 . . . . . 6 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
245, 23bd0 10274 . . . . 5 BOUNDED𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)
2524bdcab 10299 . . . 4 BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
26 df-int 3641 . . . 4 {𝑥 ∣ (𝑥𝑦𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
2725, 26bdceqir 10294 . . 3 BOUNDED {𝑥 ∣ (𝑥𝑦𝜑)}
28 bdcriota.ex . . . . 5 ∃!𝑥𝑦 𝜑
29 df-reu 2328 . . . . 5 (∃!𝑥𝑦 𝜑 ↔ ∃!𝑥(𝑥𝑦𝜑))
3028, 29mpbi 137 . . . 4 ∃!𝑥(𝑥𝑦𝜑)
31 iotaint 4905 . . . 4 (∃!𝑥(𝑥𝑦𝜑) → (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)})
3230, 31ax-mp 7 . . 3 (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)}
3327, 32bdceqir 10294 . 2 BOUNDED (℩𝑥(𝑥𝑦𝜑))
34 df-riota 5493 . 2 (𝑥𝑦 𝜑) = (℩𝑥(𝑥𝑦𝜑))
3533, 34bdceqir 10294 1 BOUNDED (𝑥𝑦 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1255   = wceq 1257  wcel 1407  [wsb 1659  ∃!weu 1914  {cab 2040  wral 2321  ∃!wreu 2323   cint 3640  cio 4890  crio 5492  BOUNDED wbd 10262  BOUNDED wbdc 10290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-bd0 10263  ax-bdim 10264  ax-bdal 10268  ax-bdel 10271  ax-bdsb 10272
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-reu 2328  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-sn 3406  df-pr 3407  df-uni 3606  df-int 3641  df-iota 4892  df-riota 5493  df-bdc 10291
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator