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Theorem iunpreima 32584
Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Assertion
Ref Expression
iunpreima (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunpreima
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4999 . . . . 5 ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵)
21a1i 11 . . . 4 (Fun 𝐹 → ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵))
32rabbidv 3440 . . 3 (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
4 funfn 6597 . . . 4 (Fun 𝐹𝐹 Fn dom 𝐹)
5 fncnvima2 7080 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
64, 5sylbi 217 . . 3 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
7 iunrab 5056 . . . 4 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵}
87a1i 11 . . 3 (Fun 𝐹 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
93, 6, 83eqtr4d 2784 . 2 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
10 fncnvima2 7080 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
114, 10sylbi 217 . . 3 (Fun 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
1211iuneq2d 5026 . 2 (Fun 𝐹 𝑥𝐴 (𝐹𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
139, 12eqtr4d 2777 1 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wrex 3067  {crab 3432   ciun 4995  ccnv 5687  dom cdm 5688  cima 5691  Fun wfun 6556   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  elrspunidl  33435
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