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

Theorem imaiun 5426
Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
imaiun (𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem imaiun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2594 . . . 4 (∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
2 vex 2577 . . . . . 6 𝑦 ∈ V
32elima3 4702 . . . . 5 (𝑦 ∈ (𝐴𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
43rexbii 2348 . . . 4 (∃𝑥𝐵 𝑦 ∈ (𝐴𝐶) ↔ ∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5 eliun 3688 . . . . . . 7 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
65anbi1i 439 . . . . . 6 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
7 r19.41v 2483 . . . . . 6 (∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
86, 7bitr4i 180 . . . . 5 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
98exbii 1512 . . . 4 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
101, 4, 93bitr4ri 206 . . 3 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
112elima3 4702 . . 3 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ ∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12 eliun 3688 . . 3 (𝑦 𝑥𝐵 (𝐴𝐶) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
1310, 11, 123bitr4i 205 . 2 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ 𝑦 𝑥𝐵 (𝐴𝐶))
1413eqriv 2053 1 (𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  wrex 2324  cop 3405   ciun 3684  cima 4375
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  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-op 3411  df-iun 3686  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  imauni  5427  uniqs  6194
  Copyright terms: Public domain W3C validator