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Theorem bdcriota 10003
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 9942 . . . . . . . 8 BOUNDED [𝑧 / 𝑥]𝜑
3 ax-bdel 9941 . . . . . . . 8 BOUNDED 𝑡𝑧
42, 3ax-bdim 9934 . . . . . . 7 BOUNDED ([𝑧 / 𝑥]𝜑𝑡𝑧)
54ax-bdal 9938 . . . . . 6 BOUNDED𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧)
6 df-ral 2311 . . . . . . . . 9 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
7 impexp 250 . . . . . . . . . . 11 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ (𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)))
87bicomi 123 . . . . . . . . . 10 ((𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
98albii 1359 . . . . . . . . 9 (∀𝑧(𝑧𝑦 → ([𝑧 / 𝑥]𝜑𝑡𝑧)) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
106, 9bitri 173 . . . . . . . 8 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧))
11 sban 1829 . . . . . . . . . . . 12 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ ([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑))
12 clelsb3 2142 . . . . . . . . . . . . 13 ([𝑧 / 𝑥]𝑥𝑦𝑧𝑦)
1312anbi1i 431 . . . . . . . . . . . 12 (([𝑧 / 𝑥]𝑥𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1411, 13bitri 173 . . . . . . . . . . 11 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ (𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑))
1514bicomi 123 . . . . . . . . . 10 ((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
1615imbi1i 227 . . . . . . . . 9 (((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1716albii 1359 . . . . . . . 8 (∀𝑧((𝑧𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
1810, 17bitri 173 . . . . . . 7 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧))
19 df-clab 2027 . . . . . . . . . 10 (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} ↔ [𝑧 / 𝑥](𝑥𝑦𝜑))
2019bicomi 123 . . . . . . . . 9 ([𝑧 / 𝑥](𝑥𝑦𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)})
2120imbi1i 227 . . . . . . . 8 (([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2221albii 1359 . . . . . . 7 (∀𝑧([𝑧 / 𝑥](𝑥𝑦𝜑) → 𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
2318, 22bitri 173 . . . . . 6 (∀𝑧𝑦 ([𝑧 / 𝑥]𝜑𝑡𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧))
245, 23bd0 9944 . . . . 5 BOUNDED𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)
2524bdcab 9969 . . . 4 BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
26 df-int 3616 . . . 4 {𝑥 ∣ (𝑥𝑦𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥𝑦𝜑)} → 𝑡𝑧)}
2725, 26bdceqir 9964 . . 3 BOUNDED {𝑥 ∣ (𝑥𝑦𝜑)}
28 bdcriota.ex . . . . 5 ∃!𝑥𝑦 𝜑
29 df-reu 2313 . . . . 5 (∃!𝑥𝑦 𝜑 ↔ ∃!𝑥(𝑥𝑦𝜑))
3028, 29mpbi 133 . . . 4 ∃!𝑥(𝑥𝑦𝜑)
31 iotaint 4880 . . . 4 (∃!𝑥(𝑥𝑦𝜑) → (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)})
3230, 31ax-mp 7 . . 3 (℩𝑥(𝑥𝑦𝜑)) = {𝑥 ∣ (𝑥𝑦𝜑)}
3327, 32bdceqir 9964 . 2 BOUNDED (℩𝑥(𝑥𝑦𝜑))
34 df-riota 5468 . 2 (𝑥𝑦 𝜑) = (℩𝑥(𝑥𝑦𝜑))
3533, 34bdceqir 9964 1 BOUNDED (𝑥𝑦 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241   = wceq 1243  wcel 1393  [wsb 1645  ∃!weu 1900  {cab 2026  wral 2306  ∃!wreu 2308   cint 3615  cio 4865  crio 5467  BOUNDED wbd 9932  BOUNDED wbdc 9960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdim 9934  ax-bdal 9938  ax-bdel 9941  ax-bdsb 9942
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-iota 4867  df-riota 5468  df-bdc 9961
This theorem is referenced by: (None)
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