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Theorem iotass 5213
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(𝜑𝑥𝐴) → (℩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.
StepHypRef Expression
1 df-iota 5196 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 unieq 3833 . . . . . . . 8 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 vex 2755 . . . . . . . . 9 𝑦 ∈ V
43unisn 3840 . . . . . . . 8 {𝑦} = 𝑦
52, 4eqtrdi 2238 . . . . . . 7 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = 𝑦)
6 df-pw 3592 . . . . . . . . . . 11 𝒫 𝐴 = {𝑥𝑥𝐴}
76sseq2i 3197 . . . . . . . . . 10 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥𝜑} ⊆ {𝑥𝑥𝐴})
8 ss2ab 3238 . . . . . . . . . 10 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
97, 8bitri 184 . . . . . . . . 9 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
109biimpri 133 . . . . . . . 8 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝒫 𝐴)
11 sspwuni 3986 . . . . . . . 8 ({𝑥𝜑} ⊆ 𝒫 𝐴 {𝑥𝜑} ⊆ 𝐴)
1210, 11sylib 122 . . . . . . 7 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝐴)
13 sseq1 3193 . . . . . . . 8 ( {𝑥𝜑} = 𝑦 → ( {𝑥𝜑} ⊆ 𝐴𝑦𝐴))
1413biimpa 296 . . . . . . 7 (( {𝑥𝜑} = 𝑦 {𝑥𝜑} ⊆ 𝐴) → 𝑦𝐴)
155, 12, 14syl2anr 290 . . . . . 6 ((∀𝑥(𝜑𝑥𝐴) ∧ {𝑥𝜑} = {𝑦}) → 𝑦𝐴)
1615ex 115 . . . . 5 (∀𝑥(𝜑𝑥𝐴) → ({𝑥𝜑} = {𝑦} → 𝑦𝐴))
1716ss2abdv 3243 . . . 4 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ {𝑦𝑦𝐴})
18 df-pw 3592 . . . 4 𝒫 𝐴 = {𝑦𝑦𝐴}
1917, 18sseqtrrdi 3219 . . 3 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20 sspwuni 3986 . . 3 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
2119, 20sylib 122 . 2 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
221, 21eqsstrid 3216 1 (∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  {cab 2175  wss 3144  𝒫 cpw 3590  {csn 3607   cuni 3824  cio 5194
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5196
This theorem is referenced by:  iotaexab  5214  fvss  5548  riotaexg  5855
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