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Theorem iotass 5105
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 5088 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 unieq 3745 . . . . . . . 8 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 vex 2689 . . . . . . . . 9 𝑦 ∈ V
43unisn 3752 . . . . . . . 8 {𝑦} = 𝑦
52, 4syl6eq 2188 . . . . . . 7 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = 𝑦)
6 df-pw 3512 . . . . . . . . . . 11 𝒫 𝐴 = {𝑥𝑥𝐴}
76sseq2i 3124 . . . . . . . . . 10 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥𝜑} ⊆ {𝑥𝑥𝐴})
8 ss2ab 3165 . . . . . . . . . 10 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
97, 8bitri 183 . . . . . . . . 9 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
109biimpri 132 . . . . . . . 8 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝒫 𝐴)
11 sspwuni 3897 . . . . . . . 8 ({𝑥𝜑} ⊆ 𝒫 𝐴 {𝑥𝜑} ⊆ 𝐴)
1210, 11sylib 121 . . . . . . 7 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝐴)
13 sseq1 3120 . . . . . . . 8 ( {𝑥𝜑} = 𝑦 → ( {𝑥𝜑} ⊆ 𝐴𝑦𝐴))
1413biimpa 294 . . . . . . 7 (( {𝑥𝜑} = 𝑦 {𝑥𝜑} ⊆ 𝐴) → 𝑦𝐴)
155, 12, 14syl2anr 288 . . . . . 6 ((∀𝑥(𝜑𝑥𝐴) ∧ {𝑥𝜑} = {𝑦}) → 𝑦𝐴)
1615ex 114 . . . . 5 (∀𝑥(𝜑𝑥𝐴) → ({𝑥𝜑} = {𝑦} → 𝑦𝐴))
1716ss2abdv 3170 . . . 4 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ {𝑦𝑦𝐴})
18 df-pw 3512 . . . 4 𝒫 𝐴 = {𝑦𝑦𝐴}
1917, 18sseqtrrdi 3146 . . 3 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20 sspwuni 3897 . . 3 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
2119, 20sylib 121 . 2 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
221, 21eqsstrid 3143 1 (∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329   = wceq 1331  {cab 2125  wss 3071  𝒫 cpw 3510  {csn 3527   cuni 3736  cio 5086
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088
This theorem is referenced by:  fvss  5435  riotaexg  5734
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