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	⊢ ∃!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 9942	. . . . . . . 8 ⊢ BOUNDED [𝑧 / 𝑥]𝜑
3		ax-bdel 9941	. . . . . . . 8 ⊢ BOUNDED 𝑡 ∈ 𝑧
4	2, 3	ax-bdim 9934	. . . . . . 7 ⊢ BOUNDED ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
5	4	ax-bdal 9938	. . . . . 6 ⊢ BOUNDED ∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)
6		df-ral 2311	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
7		impexp 250	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)))
8	7	bicomi 123	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
9	8	albii 1359	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧)) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
10	6, 9	bitri 173	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧))
11		sban 1829	. . . . . . . . . . . 12 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
12		clelsb3 2142	. . . . . . . . . . . . 13 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)
13	12	anbi1i 431	. . . . . . . . . . . 12 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
14	11, 13	bitri 173	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑))
15	14	bicomi 123	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
16	15	imbi1i 227	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
17	16	albii 1359	. . . . . . . 8 ⊢ (∀𝑧((𝑧 ∈ 𝑦 ∧ [𝑧 / 𝑥]𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
18	10, 17	bitri 173	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧))
19		df-clab 2027	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} ↔ [𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑))
20	19	bicomi 123	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) ↔ 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)})
21	20	imbi1i 227	. . . . . . . 8 ⊢ (([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
22	21	albii 1359	. . . . . . 7 ⊢ (∀𝑧([𝑧 / 𝑥](𝑥 ∈ 𝑦 ∧ 𝜑) → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
23	18, 22	bitri 173	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ([𝑧 / 𝑥]𝜑 → 𝑡 ∈ 𝑧) ↔ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧))
24	5, 23	bd0 9944	. . . . 5 ⊢ BOUNDED ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)
25	24	bdcab 9969	. . . 4 ⊢ BOUNDED {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
26		df-int 3616	. . . 4 ⊢ ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} = {𝑡 ∣ ∀𝑧(𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)} → 𝑡 ∈ 𝑧)}
27	25, 26	bdceqir 9964	. . 3 ⊢ BOUNDED ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)}
28		bdcriota.ex	. . . . 5 ⊢ ∃!𝑥 ∈ 𝑦 𝜑
29		df-reu 2313	. . . . 5 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
30	28, 29	mpbi 133	. . . 4 ⊢ ∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)
31		iotaint 4880	. . . 4 ⊢ (∃!𝑥(𝑥 ∈ 𝑦 ∧ 𝜑) → (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)) = ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)})
32	30, 31	ax-mp 7	. . 3 ⊢ (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)) = ∩ {𝑥 ∣ (𝑥 ∈ 𝑦 ∧ 𝜑)}
33	27, 32	bdceqir 9964	. 2 ⊢ BOUNDED (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
34		df-riota 5468	. 2 ⊢ (℩𝑥 ∈ 𝑦 𝜑) = (℩𝑥(𝑥 ∈ 𝑦 ∧ 𝜑))
35	33, 34	bdceqir 9964	1 ⊢ BOUNDED (℩𝑥 ∈ 𝑦 𝜑)

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)