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Theorem bdcriota 10832
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 10771 . . . . . . . 8 BOUNDED [𝑧 / 𝑥]𝜑
3 ax-bdel 10770 . . . . . . . 8 BOUNDED 𝑡𝑧
42, 3ax-bdim 10763 . . . . . . 7 BOUNDED ([𝑧 / 𝑥]𝜑𝑡𝑧)
54ax-bdal 10767 . . . . . 6 BOUNDED𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧)
6 df-ral 2354 . . . . . . . . 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 1400 . . . . . . . . 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 1871 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ ([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑))
12 clelsb3 2184 . . . . . . . . . . . . 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 1400 . . . . . . . 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 2069 . . . . . . . . . 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 1400 . . . . . . 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 10773 . . . . 5 BOUNDED𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)
2524bdcab 10798 . . . 4 BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
26 df-int 3645 . . . 4 {𝑥 ∣ (𝑥𝑦𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
2725, 26bdceqir 10793 . . 3 BOUNDED {𝑥 ∣ (𝑥𝑦𝜑)}
28 bdcriota.ex . . . . 5 ∃!𝑥𝑦 𝜑
29 df-reu 2356 . . . . 5 (∃!𝑥𝑦 𝜑 ↔ ∃!𝑥(𝑥𝑦𝜑))
3028, 29mpbi 143 . . . 4 ∃!𝑥(𝑥𝑦𝜑)
31 iotaint 4910 . . . 4 (∃!𝑥(𝑥𝑦𝜑) → (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)})
3230, 31ax-mp 7 . . 3 (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)}
3327, 32bdceqir 10793 . 2 BOUNDED (℩𝑥(𝑥𝑦𝜑))
34 df-riota 5499 . 2 (𝑥𝑦 𝜑) = (℩𝑥(𝑥𝑦𝜑))
3533, 34bdceqir 10793 1 BOUNDED (𝑥𝑦 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wcel 1434  [wsb 1686  ∃!weu 1942  {cab 2068  wral 2349  ∃!wreu 2351   cint 3644  cio 4895  crio 5498  BOUNDED wbd 10761  BOUNDED wbdc 10789
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bd0 10762  ax-bdim 10763  ax-bdal 10767  ax-bdel 10770  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-iota 4897  df-riota 5499  df-bdc 10790
This theorem is referenced by: (None)
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