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Theorem imaiun 7193
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 3270 . . . 4 (∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
2 vex 3448 . . . . . 6 𝑦 ∈ V
32elima3 6021 . . . . 5 (𝑦 ∈ (𝐴𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
43rexbii 3094 . . . 4 (∃𝑥𝐵 𝑦 ∈ (𝐴𝐶) ↔ ∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5 eliun 4959 . . . . . . 7 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
65anbi1i 625 . . . . . 6 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
7 r19.41v 3182 . . . . . 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 1851 . . . 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 6021 . . 3 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ ∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12 eliun 4959 . . 3 (𝑦 𝑥𝐵 (𝐴𝐶) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
1310, 11, 123bitr4i 303 . 2 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ 𝑦 𝑥𝐵 (𝐴𝐶))
1413eqriv 2730 1 (𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3070  cop 4593   ciun 4955  cima 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4957  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647
This theorem is referenced by:  imauni  7194  uniqs  8719  hsmexlem4  10370  hsmexlem5  10371  xkococnlem  23026  ismbf3d  25034  mbfimaopnlem  25035  i1fima  25058  i1fd  25061  itg1addlem5  25081  limciun  25274  sibfof  32997  eulerpartlemgh  33035  poimirlem30  36154  itg2addnclem2  36176  ftc1anclem6  36202  uniqsALTV  36836  smfresal  45115
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