ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iotass Structured version   GIF version

Theorem iotass 4827
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 4810 . 2 (℩xφ) = {y ∣ {xφ} = {y}}
2 unieq 3580 . . . . . . . 8 ({xφ} = {y} → {xφ} = {y})
3 vex 2554 . . . . . . . . 9 y V
43unisn 3587 . . . . . . . 8 {y} = y
52, 4syl6eq 2085 . . . . . . 7 ({xφ} = {y} → {xφ} = y)
6 df-pw 3353 . . . . . . . . . . 11 𝒫 A = {xxA}
76sseq2i 2964 . . . . . . . . . 10 ({xφ} ⊆ 𝒫 A ↔ {xφ} ⊆ {xxA})
8 ss2ab 3002 . . . . . . . . . 10 ({xφ} ⊆ {xxA} ↔ x(φxA))
97, 8bitri 173 . . . . . . . . 9 ({xφ} ⊆ 𝒫 Ax(φxA))
109biimpri 124 . . . . . . . 8 (x(φxA) → {xφ} ⊆ 𝒫 A)
11 sspwuni 3730 . . . . . . . 8 ({xφ} ⊆ 𝒫 A {xφ} ⊆ A)
1210, 11sylib 127 . . . . . . 7 (x(φxA) → {xφ} ⊆ A)
13 sseq1 2960 . . . . . . . 8 ( {xφ} = y → ( {xφ} ⊆ AyA))
1413biimpa 280 . . . . . . 7 (( {xφ} = y {xφ} ⊆ A) → yA)
155, 12, 14syl2anr 274 . . . . . 6 ((x(φxA) {xφ} = {y}) → yA)
1615ex 108 . . . . 5 (x(φxA) → ({xφ} = {y} → yA))
1716ss2abdv 3007 . . . 4 (x(φxA) → {y ∣ {xφ} = {y}} ⊆ {yyA})
18 df-pw 3353 . . . 4 𝒫 A = {yyA}
1917, 18syl6sseqr 2986 . . 3 (x(φxA) → {y ∣ {xφ} = {y}} ⊆ 𝒫 A)
20 sspwuni 3730 . . 3 ({y ∣ {xφ} = {y}} ⊆ 𝒫 A {y ∣ {xφ} = {y}} ⊆ A)
2119, 20sylib 127 . 2 (x(φxA) → {y ∣ {xφ} = {y}} ⊆ A)
221, 21syl5eqss 2983 1 (x(φxA) → (℩xφ) ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242  {cab 2023  wss 2911  𝒫 cpw 3351  {csn 3367   cuni 3571  cio 4808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by:  fvss  5132  riotaexg  5415
  Copyright terms: Public domain W3C validator