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Theorem iunpreima 29721
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 4659 . . . . 5 ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵)
21a1i 11 . . . 4 (Fun 𝐹 → ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵))
32rabbidv 3339 . . 3 (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
4 funfn 6060 . . . 4 (Fun 𝐹𝐹 Fn dom 𝐹)
5 fncnvima2 6484 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
64, 5sylbi 207 . . 3 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵})
7 iunrab 4702 . . . 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 2815 . 2 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
10 fncnvima2 6484 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
114, 10sylbi 207 . . 3 (Fun 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
1211iuneq2d 4682 . 2 (Fun 𝐹 𝑥𝐴 (𝐹𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
139, 12eqtr4d 2808 1 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wrex 3062  {crab 3065   ciun 4655  ccnv 5249  dom cdm 5250  cima 5253  Fun wfun 6024   Fn wfn 6025  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038
This theorem is referenced by: (None)
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