Theorem iotass 5350

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 5332	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		unieq 3939	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 2824	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	unisn 3946	. . . . . . . 8 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	eqtrdi 2287	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = 𝑦)
6		df-pw 3687	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	6	sseq2i 3275	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴})
8		ss2ab 3316	. . . . . . . . . 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 4092	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
12	10, 11	sylib 122	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
13		sseq1 3271	. . . . . . . 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 3321	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ {𝑦 ∣ 𝑦 ⊆ 𝐴})
18		df-pw 3687	. . . 4 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
19	17, 18	sseqtrrdi 3297	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20		sspwuni 4092	. . 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 3294	1 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1400   = wceq 1402  {cab 2224   ⊆ wss 3220  𝒫 cpw 3685  {csn 3705  ∪ cuni 3930  ℩cio 5330

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-iota 5332

This theorem is referenced by:  iotaexab  5351  fvss  5704  riotaexg  6032