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Theorem bdcriota 11431
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 11370 . . . . . . . 8 BOUNDED [𝑧 / 𝑥]𝜑
3 ax-bdel 11369 . . . . . . . 8 BOUNDED 𝑡𝑧
42, 3ax-bdim 11362 . . . . . . 7 BOUNDED ([𝑧 / 𝑥]𝜑𝑡𝑧)
54ax-bdal 11366 . . . . . 6 BOUNDED𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧)
6 df-ral 2364 . . . . . . . . 9 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
7 impexp 259 . . . . . . . . . . 11 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ (𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
87bicomi 130 . . . . . . . . . 10 ((𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
98albii 1404 . . . . . . . . 9 (∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
106, 9bitri 182 . . . . . . . 8 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
11 sban 1877 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ ([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑))
12 clelsb3 2192 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑥𝑦𝑧𝑦)
1312anbi1i 446 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1411, 13bitri 182 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1514bicomi 130 . . . . . . . . . 10 ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
1615imbi1i 236 . . . . . . . . 9 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1716albii 1404 . . . . . . . 8 (∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1810, 17bitri 182 . . . . . . 7 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
19 df-clab 2075 . . . . . . . . . 10 (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
2019bicomi 130 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)})
2120imbi1i 236 . . . . . . . 8 (([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2221albii 1404 . . . . . . 7 (∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2318, 22bitri 182 . . . . . 6 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
245, 23bd0 11372 . . . . 5 BOUNDED𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)
2524bdcab 11397 . . . 4 BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
26 df-int 3684 . . . 4 {𝑥 ∣ (𝑥𝑦𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
2725, 26bdceqir 11392 . . 3 BOUNDED {𝑥 ∣ (𝑥𝑦𝜑)}
28 bdcriota.ex . . . . 5 ∃!𝑥𝑦 𝜑
29 df-reu 2366 . . . . 5 (∃!𝑥𝑦 𝜑 ↔ ∃!𝑥(𝑥𝑦𝜑))
3028, 29mpbi 143 . . . 4 ∃!𝑥(𝑥𝑦𝜑)
31 iotaint 4980 . . . 4 (∃!𝑥(𝑥𝑦𝜑) → (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)})
3230, 31ax-mp 7 . . 3 (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)}
3327, 32bdceqir 11392 . 2 BOUNDED (℩𝑥(𝑥𝑦𝜑))
34 df-riota 5590 . 2 (𝑥𝑦 𝜑) = (℩𝑥(𝑥𝑦𝜑))
3533, 34bdceqir 11392 1 BOUNDED (𝑥𝑦 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287   = wceq 1289  wcel 1438  [wsb 1692  ∃!weu 1948  {cab 2074  wral 2359  ∃!wreu 2361   cint 3683  cio 4965  crio 5589  BOUNDED wbd 11360  BOUNDED wbdc 11388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11361  ax-bdim 11362  ax-bdal 11366  ax-bdel 11369  ax-bdsb 11370
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-iota 4967  df-riota 5590  df-bdc 11389
This theorem is referenced by: (None)
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