Theorem iotass 4827

Description: Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)

Assertion

Ref	Expression
iotass	⊢ (∀x(φ → x ⊆ A) → (℩xφ) ⊆ A)

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem iotass

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 4810	. 2 ⊢ (℩xφ) = ∪ {y ∣ {x ∣ φ} = {y}}
2		unieq 3580	. . . . . . . 8 ⊢ ({x ∣ φ} = {y} → ∪ {x ∣ φ} = ∪ {y})
3		vex 2554	. . . . . . . . 9 ⊢ y ∈ V
4	3	unisn 3587	. . . . . . . 8 ⊢ ∪ {y} = y
5	2, 4	syl6eq 2085	. . . . . . 7 ⊢ ({x ∣ φ} = {y} → ∪ {x ∣ φ} = y)
6		df-pw 3353	. . . . . . . . . . 11 ⊢ 𝒫 A = {x ∣ x ⊆ A}
7	6	sseq2i 2964	. . . . . . . . . 10 ⊢ ({x ∣ φ} ⊆ 𝒫 A ↔ {x ∣ φ} ⊆ {x ∣ x ⊆ A})
8		ss2ab 3002	. . . . . . . . . 10 ⊢ ({x ∣ φ} ⊆ {x ∣ x ⊆ A} ↔ ∀x(φ → x ⊆ A))
9	7, 8	bitri 173	. . . . . . . . 9 ⊢ ({x ∣ φ} ⊆ 𝒫 A ↔ ∀x(φ → x ⊆ A))
10	9	biimpri 124	. . . . . . . 8 ⊢ (∀x(φ → x ⊆ A) → {x ∣ φ} ⊆ 𝒫 A)
11		sspwuni 3730	. . . . . . . 8 ⊢ ({x ∣ φ} ⊆ 𝒫 A ↔ ∪ {x ∣ φ} ⊆ A)
12	10, 11	sylib 127	. . . . . . 7 ⊢ (∀x(φ → x ⊆ A) → ∪ {x ∣ φ} ⊆ A)
13		sseq1 2960	. . . . . . . 8 ⊢ (∪ {x ∣ φ} = y → (∪ {x ∣ φ} ⊆ A ↔ y ⊆ A))
14	13	biimpa 280	. . . . . . 7 ⊢ ((∪ {x ∣ φ} = y ∧ ∪ {x ∣ φ} ⊆ A) → y ⊆ A)
15	5, 12, 14	syl2anr 274	. . . . . 6 ⊢ ((∀x(φ → x ⊆ A) ∧ {x ∣ φ} = {y}) → y ⊆ A)
16	15	ex 108	. . . . 5 ⊢ (∀x(φ → x ⊆ A) → ({x ∣ φ} = {y} → y ⊆ A))
17	16	ss2abdv 3007	. . . 4 ⊢ (∀x(φ → x ⊆ A) → {y ∣ {x ∣ φ} = {y}} ⊆ {y ∣ y ⊆ A})
18		df-pw 3353	. . . 4 ⊢ 𝒫 A = {y ∣ y ⊆ A}
19	17, 18	syl6sseqr 2986	. . 3 ⊢ (∀x(φ → x ⊆ A) → {y ∣ {x ∣ φ} = {y}} ⊆ 𝒫 A)
20		sspwuni 3730	. . 3 ⊢ ({y ∣ {x ∣ φ} = {y}} ⊆ 𝒫 A ↔ ∪ {y ∣ {x ∣ φ} = {y}} ⊆ A)
21	19, 20	sylib 127	. 2 ⊢ (∀x(φ → x ⊆ A) → ∪ {y ∣ {x ∣ φ} = {y}} ⊆ A)
22	1, 21	syl5eqss 2983	1 ⊢ (∀x(φ → x ⊆ A) → (℩xφ) ⊆ A)

Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242  {cab 2023   ⊆ wss 2911  𝒫 cpw 3351  {csn 3367  ∪ cuni 3571  ℩cio 4808

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810

This theorem is referenced by:  fvss  5132  riotaexg  5415