Theorem iotass 5311

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 5293	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		unieq 3907	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 2806	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	unisn 3914	. . . . . . . 8 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	eqtrdi 2280	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = 𝑦)
6		df-pw 3658	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	6	sseq2i 3255	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴})
8		ss2ab 3296	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
9	7, 8	bitri 184	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ⊆ 𝐴))
10	9	biimpri 133	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴)
11		sspwuni 4060	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
12	10, 11	sylib 122	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
13		sseq1 3251	. . . . . . . 8 ⊢ (∪ {𝑥 ∣ 𝜑} = 𝑦 → (∪ {𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
14	13	biimpa 296	. . . . . . 7 ⊢ ((∪ {𝑥 ∣ 𝜑} = 𝑦 ∧ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
15	5, 12, 14	syl2anr 290	. . . . . 6 ⊢ ((∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) ∧ {𝑥 ∣ 𝜑} = {𝑦}) → 𝑦 ⊆ 𝐴)
16	15	ex 115	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 ⊆ 𝐴))
17	16	ss2abdv 3301	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ {𝑦 ∣ 𝑦 ⊆ 𝐴})
18		df-pw 3658	. . . 4 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
19	17, 18	sseqtrrdi 3277	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20		sspwuni 4060	. . 3 ⊢ ({𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴 ↔ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
21	19, 20	sylib 122	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝐴)
22	1, 21	eqsstrid 3274	1 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1396   = wceq 1398  {cab 2217   ⊆ wss 3201  𝒫 cpw 3656  {csn 3673  ∪ cuni 3898  ℩cio 5291

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293

This theorem is referenced by:  iotaexab  5312  fvss  5662  riotaexg  5985