Theorem bdcriota 13252

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	⊢ ∃!𝑥 ∈ 𝑦 𝜑

Assertion

Ref	Expression
bdcriota	⊢ BOUNDED (℩𝑥 ∈ 𝑦 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.

Step	Hyp	Ref	Expression
1		bdcriota.bd	. . . . . . . . 9 ⊢ BOUNDED 𝜑
2	1	ax-bdsb 13191	. . . . . . . 8 ⊢ BOUNDED [𝑧 / 𝑥]𝜑
3		ax-bdel 13190	. . . . . . . 8 ⊢ BOUNDED 𝑡 ∈ 𝑧
4	2, 3	ax-bdim 13183	. . . . . . 7 ⊢ BOUNDED ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
5	4	ax-bdal 13187	. . . . . 6 ⊢ BOUNDED ∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
6		df-ral 2422	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
7		impexp 261	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
8	7	bicomi 131	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
9	8	albii 1447	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
10	6, 9	bitri 183	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
11		sban 1929	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
12		clelsb3 2245	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)
13	12	anbi1i 454	. . . . . . . . . . . 12 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
14	11, 13	bitri 183	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
15	14	bicomi 131	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
16	15	imbi1i 237	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
17	16	albii 1447	. . . . . . . 8 ⊢ (∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
18	10, 17	bitri 183	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
19		df-clab 2127	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
20	19	bicomi 131	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)})
21	20	imbi1i 237	. . . . . . . 8 ⊢ (([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
22	21	albii 1447	. . . . . . 7 ⊢ (∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
23	18, 22	bitri 183	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
24	5, 23	bd0 13193	. . . . 5 ⊢ BOUNDED ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)
25	24	bdcab 13218	. . . 4 ⊢ BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
26		df-int 3780	. . . 4 ⊢ ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
27	25, 26	bdceqir 13213	. . 3 ⊢ BOUNDED ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)}
28		bdcriota.ex	. . . . 5 ⊢ ∃!𝑥 ∈ 𝑦 𝜑
29		df-reu 2424	. . . . 5 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
30	28, 29	mpbi 144	. . . 4 ⊢ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)
31		iotaint 5109	. . . 4 ⊢ (∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)) = ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)})
32	30, 31	ax-mp 5	. . 3 ⊢ (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)) = ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)}
33	27, 32	bdceqir 13213	. 2 ⊢ BOUNDED (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
34		df-riota 5738	. 2 ⊢ (℩𝑥 ∈ 𝑦 𝜑) = (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
35	33, 34	bdceqir 13213	1 ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332   ∈ wcel 1481  [wsb 1736  ∃!weu 2000  {cab 2126  ∀wral 2417  ∃!wreu 2419  ∩ cint 3779  ℩cio 5094  ℩crio 5737  BOUNDED wbd 13181  BOUNDED wbdc 13209

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdim 13183  ax-bdal 13187  ax-bdel 13190  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-iota 5096  df-riota 5738  df-bdc 13210

This theorem is referenced by: (None)