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Theorem iotass 5195
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 5178 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 unieq 3818 . . . . . . . 8 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 vex 2740 . . . . . . . . 9 𝑦 ∈ V
43unisn 3825 . . . . . . . 8 {𝑦} = 𝑦
52, 4eqtrdi 2226 . . . . . . 7 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = 𝑦)
6 df-pw 3577 . . . . . . . . . . 11 𝒫 𝐴 = {𝑥𝑥𝐴}
76sseq2i 3182 . . . . . . . . . 10 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥𝜑} ⊆ {𝑥𝑥𝐴})
8 ss2ab 3223 . . . . . . . . . 10 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
97, 8bitri 184 . . . . . . . . 9 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
109biimpri 133 . . . . . . . 8 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝒫 𝐴)
11 sspwuni 3971 . . . . . . . 8 ({𝑥𝜑} ⊆ 𝒫 𝐴 {𝑥𝜑} ⊆ 𝐴)
1210, 11sylib 122 . . . . . . 7 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝐴)
13 sseq1 3178 . . . . . . . 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 3228 . . . 4 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ {𝑦𝑦𝐴})
18 df-pw 3577 . . . 4 𝒫 𝐴 = {𝑦𝑦𝐴}
1917, 18sseqtrrdi 3204 . . 3 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20 sspwuni 3971 . . 3 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
2119, 20sylib 122 . 2 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
221, 21eqsstrid 3201 1 (∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351   = wceq 1353  {cab 2163  wss 3129  𝒫 cpw 3575  {csn 3592   cuni 3809  cio 5176
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-iota 5178
This theorem is referenced by:  fvss  5529  riotaexg  5834
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