Theorem iotass 5100

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 5083	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		unieq 3740	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		vex 2684	. . . . . . . . 9 ⊢ 𝑦 ∈ V
4	3	unisn 3747	. . . . . . . 8 ⊢ ∪ {𝑦} = 𝑦
5	2, 4	syl6eq 2186	. . . . . . 7 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = 𝑦)
6		df-pw 3507	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
7	6	sseq2i 3119	. . . . . . . . . 10 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴})
8		ss2ab 3160	. . . . . . . . . 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 3892	. . . . . . . 8 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
12	10, 11	sylib 121	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → ∪ {𝑥 ∣ 𝜑} ⊆ 𝐴)
13		sseq1 3115	. . . . . . . 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 3165	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ {𝑦 ∣ 𝑦 ⊆ 𝐴})
18		df-pw 3507	. . . 4 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴}
19	17, 18	sseqtrrdi 3141	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20		sspwuni 3892	. . 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 3138	1 ⊢ (∀𝑥(𝜑 → 𝑥 ⊆ 𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331  {cab 2123   ⊆ wss 3066  𝒫 cpw 3505  {csn 3522  ∪ cuni 3731  ℩cio 5081

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083

This theorem is referenced by:  fvss  5428  riotaexg  5727