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Theorem iotass 4911
 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 4894 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 unieq 3616 . . . . . . . 8 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = {𝑦})
3 vex 2577 . . . . . . . . 9 𝑦 ∈ V
43unisn 3623 . . . . . . . 8 {𝑦} = 𝑦
52, 4syl6eq 2104 . . . . . . 7 ({𝑥𝜑} = {𝑦} → {𝑥𝜑} = 𝑦)
6 df-pw 3388 . . . . . . . . . . 11 𝒫 𝐴 = {𝑥𝑥𝐴}
76sseq2i 2997 . . . . . . . . . 10 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ {𝑥𝜑} ⊆ {𝑥𝑥𝐴})
8 ss2ab 3035 . . . . . . . . . 10 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
97, 8bitri 177 . . . . . . . . 9 ({𝑥𝜑} ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
109biimpri 128 . . . . . . . 8 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝒫 𝐴)
11 sspwuni 3766 . . . . . . . 8 ({𝑥𝜑} ⊆ 𝒫 𝐴 {𝑥𝜑} ⊆ 𝐴)
1210, 11sylib 131 . . . . . . 7 (∀𝑥(𝜑𝑥𝐴) → {𝑥𝜑} ⊆ 𝐴)
13 sseq1 2993 . . . . . . . 8 ( {𝑥𝜑} = 𝑦 → ( {𝑥𝜑} ⊆ 𝐴𝑦𝐴))
1413biimpa 284 . . . . . . 7 (( {𝑥𝜑} = 𝑦 {𝑥𝜑} ⊆ 𝐴) → 𝑦𝐴)
155, 12, 14syl2anr 278 . . . . . 6 ((∀𝑥(𝜑𝑥𝐴) ∧ {𝑥𝜑} = {𝑦}) → 𝑦𝐴)
1615ex 112 . . . . 5 (∀𝑥(𝜑𝑥𝐴) → ({𝑥𝜑} = {𝑦} → 𝑦𝐴))
1716ss2abdv 3040 . . . 4 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ {𝑦𝑦𝐴})
18 df-pw 3388 . . . 4 𝒫 𝐴 = {𝑦𝑦𝐴}
1917, 18syl6sseqr 3019 . . 3 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴)
20 sspwuni 3766 . . 3 ({𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝒫 𝐴 {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
2119, 20sylib 131 . 2 (∀𝑥(𝜑𝑥𝐴) → {𝑦 ∣ {𝑥𝜑} = {𝑦}} ⊆ 𝐴)
221, 21syl5eqss 3016 1 (∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257   = wceq 1259  {cab 2042   ⊆ wss 2944  𝒫 cpw 3386  {csn 3402  ∪ cuni 3607  ℩cio 4892 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-iota 4894 This theorem is referenced by:  fvss  5216  riotaexg  5499
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