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Theorem iotass 4984
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 4967 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 unieq 3657 . . . . . . . 8 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 vex 2622 . . . . . . . . 9 𝑦 ∈ V
43unisn 3664 . . . . . . . 8 {𝑦} = 𝑦
52, 4syl6eq 2136 . . . . . . 7 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = 𝑦)
6 df-pw 3427 . . . . . . . . . . 11 𝒫 𝐴 = {𝑥𝑥𝐴}
76sseq2i 3049 . . . . . . . . . 10 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥𝜑} ⊆ {𝑥𝑥𝐴})
8 ss2ab 3087 . . . . . . . . . 10 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
97, 8bitri 182 . . . . . . . . 9 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
109biimpri 131 . . . . . . . 8 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝒫 𝐴)
11 sspwuni 3808 . . . . . . . 8 ({𝑥𝜑} ⊆ 𝒫 𝐴 {𝑥𝜑} ⊆ 𝐴)
1210, 11sylib 120 . . . . . . 7 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝐴)
13 sseq1 3045 . . . . . . . 8 ( {𝑥𝜑} = 𝑦 → ( {𝑥𝜑} ⊆ 𝐴𝑦𝐴))
1413biimpa 290 . . . . . . 7 (( {𝑥𝜑} = 𝑦 {𝑥𝜑} ⊆ 𝐴) → 𝑦𝐴)
155, 12, 14syl2anr 284 . . . . . 6 ((∀𝑥(𝜑𝑥𝐴) ∧ {𝑥𝜑} = {𝑦}) → 𝑦𝐴)
1615ex 113 . . . . 5 (∀𝑥(𝜑𝑥𝐴) → ({𝑥𝜑} = {𝑦} → 𝑦𝐴))
1716ss2abdv 3092 . . . 4 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ {𝑦𝑦𝐴})
18 df-pw 3427 . . . 4 𝒫 𝐴 = {𝑦𝑦𝐴}
1917, 18syl6sseqr 3071 . . 3 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20 sspwuni 3808 . . 3 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
2119, 20sylib 120 . 2 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
221, 21syl5eqss 3068 1 (∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  {cab 2074  wss 2997  𝒫 cpw 3425  {csn 3441   cuni 3648  cio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-iota 4967
This theorem is referenced by:  fvss  5303  riotaexg  5594
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