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Theorem bdcriota 12892
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 12831 . . . . . . . 8 BOUNDED [𝑧 / 𝑥]𝜑
3 ax-bdel 12830 . . . . . . . 8 BOUNDED 𝑡𝑧
42, 3ax-bdim 12823 . . . . . . 7 BOUNDED ([𝑧 / 𝑥]𝜑𝑡𝑧)
54ax-bdal 12827 . . . . . 6 BOUNDED𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧)
6 df-ral 2396 . . . . . . . . 9 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
7 impexp 261 . . . . . . . . . . 11 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ (𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
87bicomi 131 . . . . . . . . . 10 ((𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
98albii 1429 . . . . . . . . 9 (∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
106, 9bitri 183 . . . . . . . 8 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
11 sban 1904 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ ([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑))
12 clelsb3 2220 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑥𝑦𝑧𝑦)
1312anbi1i 451 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1411, 13bitri 183 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1514bicomi 131 . . . . . . . . . 10 ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
1615imbi1i 237 . . . . . . . . 9 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1716albii 1429 . . . . . . . 8 (∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1810, 17bitri 183 . . . . . . 7 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
19 df-clab 2102 . . . . . . . . . 10 (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
2019bicomi 131 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)})
2120imbi1i 237 . . . . . . . 8 (([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2221albii 1429 . . . . . . 7 (∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2318, 22bitri 183 . . . . . 6 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
245, 23bd0 12833 . . . . 5 BOUNDED𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)
2524bdcab 12858 . . . 4 BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
26 df-int 3740 . . . 4 {𝑥 ∣ (𝑥𝑦𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
2725, 26bdceqir 12853 . . 3 BOUNDED {𝑥 ∣ (𝑥𝑦𝜑)}
28 bdcriota.ex . . . . 5 ∃!𝑥𝑦 𝜑
29 df-reu 2398 . . . . 5 (∃!𝑥𝑦 𝜑 ↔ ∃!𝑥(𝑥𝑦𝜑))
3028, 29mpbi 144 . . . 4 ∃!𝑥(𝑥𝑦𝜑)
31 iotaint 5069 . . . 4 (∃!𝑥(𝑥𝑦𝜑) → (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)})
3230, 31ax-mp 5 . . 3 (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)}
3327, 32bdceqir 12853 . 2 BOUNDED (℩𝑥(𝑥𝑦𝜑))
34 df-riota 5696 . 2 (𝑥𝑦 𝜑) = (℩𝑥(𝑥𝑦𝜑))
3533, 34bdceqir 12853 1 BOUNDED (𝑥𝑦 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wcel 1463  [wsb 1718  ∃!weu 1975  {cab 2101  wral 2391  ∃!wreu 2393   cint 3739  cio 5054  crio 5695  BOUNDED wbd 12821  BOUNDED wbdc 12849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12822  ax-bdim 12823  ax-bdal 12827  ax-bdel 12830  ax-bdsb 12831
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-iota 5056  df-riota 5696  df-bdc 12850
This theorem is referenced by: (None)
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