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Theorem iunpreima 32577
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 4995 . . . . 5 ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵)
21a1i 11 . . . 4 (Fun 𝐹 → ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵))
32rabbidv 3444 . . 3 (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
4 funfn 6596 . . . 4 (Fun 𝐹𝐹 Fn dom 𝐹)
5 fncnvima2 7081 . . . 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 5052 . . . 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 2787 . 2 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
10 fncnvima2 7081 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
114, 10sylbi 217 . . 3 (Fun 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
1211iuneq2d 5022 . 2 (Fun 𝐹 𝑥𝐴 (𝐹𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
139, 12eqtr4d 2780 1 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  {crab 3436   ciun 4991  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
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-10 2141  ax-11 2157  ax-12 2177  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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  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-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  elrspunidl  33456
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