Theorem bdcriota 16670

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 16609	. . . . . . . 8 ⊢ BOUNDED [𝑧 / 𝑥]𝜑
3		ax-bdel 16608	. . . . . . . 8 ⊢ BOUNDED 𝑡 ∈ 𝑧
4	2, 3	ax-bdim 16601	. . . . . . 7 ⊢ BOUNDED ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
5	4	ax-bdal 16605	. . . . . 6 ⊢ BOUNDED ∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
6		df-ral 2527	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
7		impexp 263	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
8	7	bicomi 132	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
9	8	albii 1519	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
10	6, 9	bitri 184	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
11		sban 2011	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
12		clelsb1 2339	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)
13	12	anbi1i 458	. . . . . . . . . . . 12 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
14	11, 13	bitri 184	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
15	14	bicomi 132	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
16	15	imbi1i 238	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
17	16	albii 1519	. . . . . . . 8 ⊢ (∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
18	10, 17	bitri 184	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
19		df-clab 2221	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
20	19	bicomi 132	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)})
21	20	imbi1i 238	. . . . . . . 8 ⊢ (([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
22	21	albii 1519	. . . . . . 7 ⊢ (∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
23	18, 22	bitri 184	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
24	5, 23	bd0 16611	. . . . 5 ⊢ BOUNDED ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)
25	24	bdcab 16636	. . . 4 ⊢ BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
26		df-int 3952	. . . 4 ⊢ ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
27	25, 26	bdceqir 16631	. . 3 ⊢ BOUNDED ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)}
28		bdcriota.ex	. . . . 5 ⊢ ∃!𝑥 ∈ 𝑦 𝜑
29		df-reu 2529	. . . . 5 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
30	28, 29	mpbi 145	. . . 4 ⊢ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)
31		iotaint 5328	. . . 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 16631	. 2 ⊢ BOUNDED (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
34		df-riota 6005	. 2 ⊢ (℩𝑥 ∈ 𝑦 𝜑) = (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
35	33, 34	bdceqir 16631	1 ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398  [wsb 1811  ∃!weu 2082   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃!wreu 2524  ∩ cint 3951  ℩cio 5312  ℩crio 6004  BOUNDED wbd 16599  BOUNDED wbdc 16627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-bd0 16600  ax-bdim 16601  ax-bdal 16605  ax-bdel 16608  ax-bdsb 16609

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-iota 5314  df-riota 6005  df-bdc 16628

This theorem is referenced by: (None)