Theorem iotass 5170

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 5153	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		unieq 3798	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 2729	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	unisn 3805	. . . . . . . 8 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	eqtrdi 2215	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = 𝑦)
6		df-pw 3561	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	6	sseq2i 3169	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴})
8		ss2ab 3210	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
9	7, 8	bitri 183	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
10	9	biimpri 132	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴)
11		sspwuni 3950	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
12	10, 11	sylib 121	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
13		sseq1 3165	. . . . . . . 8 ⊢ (∪ {𝑥 ∣ 𝜑} = 𝑦 → (∪ {𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
14	13	biimpa 294	. . . . . . 7 ⊢ ((∪ {𝑥 ∣ 𝜑} = 𝑦 ∧ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
15	5, 12, 14	syl2anr 288	. . . . . 6 ⊢ ((∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) ∧ {𝑥 ∣ 𝜑} = {𝑦}) → 𝑦 ⊆ 𝐴)
16	15	ex 114	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 ⊆ 𝐴))
17	16	ss2abdv 3215	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ {𝑦 ∣ 𝑦 ⊆ 𝐴})
18		df-pw 3561	. . . 4 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
19	17, 18	sseqtrrdi 3191	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20		sspwuni 3950	. . 3 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴 ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
21	19, 20	sylib 121	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
22	1, 21	eqsstrid 3188	1 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1341   = wceq 1343  {cab 2151   ⊆ wss 3116  𝒫 cpw 3559  {csn 3576  ∪ cuni 3789  ℩cio 5151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by:  fvss  5500  riotaexg  5802