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Theorem imaiun 5629
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 2683 . . . 4 (∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
2 vex 2663 . . . . . 6 𝑦 ∈ V
32elima3 4858 . . . . 5 (𝑦 ∈ (𝐴𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
43rexbii 2419 . . . 4 (∃𝑥𝐵 𝑦 ∈ (𝐴𝐶) ↔ ∃𝑥𝐵𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5 eliun 3787 . . . . . . 7 (𝑧 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑧𝐶)
65anbi1i 453 . . . . . 6 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
7 r19.41v 2564 . . . . . 6 (∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (∃𝑥𝐵 𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
86, 7bitr4i 186 . . . . 5 ((𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
98exbii 1569 . . . 4 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧𝑥𝐵 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
101, 4, 93bitr4ri 212 . . 3 (∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
112elima3 4858 . . 3 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ ∃𝑧(𝑧 𝑥𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12 eliun 3787 . . 3 (𝑦 𝑥𝐵 (𝐴𝐶) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴𝐶))
1310, 11, 123bitr4i 211 . 2 (𝑦 ∈ (𝐴 𝑥𝐵 𝐶) ↔ 𝑦 𝑥𝐵 (𝐴𝐶))
1413eqriv 2114 1 (𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wex 1453  wcel 1465  wrex 2394  cop 3500   ciun 3783  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  imauni  5630  uniqs  6455
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