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Theorem imaiun 7265
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 3288 . . . 4 (∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
2 vex 3484 . . . . . 6 𝑦 ∈ V
32elima3 6085 . . . . 5 (𝑦 ∈ (𝐴𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
43rexbii 3094 . . . 4 (∃𝑥𝐵 𝑦 ∈ (𝐴𝐶) ↔ ∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5 eliun 4995 . . . . . . 7 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
65anbi1i 624 . . . . . 6 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
7 r19.41v 3189 . . . . . 6 (∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
86, 7bitr4i 278 . . . . 5 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
98exbii 1848 . . . 4 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
101, 4, 93bitr4ri 304 . . 3 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
112elima3 6085 . . 3 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ ∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12 eliun 4995 . . 3 (𝑦 𝑥𝐵 (𝐴𝐶) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
1310, 11, 123bitr4i 303 . 2 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ 𝑦 𝑥𝐵 (𝐴𝐶))
1413eqriv 2734 1 (𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cop 4632   ciun 4991  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  imauni  7266  uniqs  8817  hsmexlem4  10469  hsmexlem5  10470  xkococnlem  23667  ismbf3d  25689  mbfimaopnlem  25690  i1fima  25713  i1fd  25716  itg1addlem5  25735  limciun  25929  sibfof  34342  eulerpartlemgh  34380  poimirlem30  37657  itg2addnclem2  37679  ftc1anclem6  37705  uniqsALTV  38330  smfresal  46803
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