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Theorem iunpreima 29509
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 4556 . . . . 5 ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵)
21a1i 11 . . . 4 (Fun 𝐹 → ((𝐹𝑦) ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵))
32rabbidv 3220 . . 3 (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝑥𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝐵})
4 funfn 5956 . . . 4 (Fun 𝐹𝐹 Fn dom 𝐹)
5 fncnvima2 6379 . . . 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 4599 . . . 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 2695 . 2 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
10 fncnvima2 6379 . . . 4 (𝐹 Fn dom 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
114, 10sylbi 207 . . 3 (Fun 𝐹 → (𝐹𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
1211iuneq2d 4579 . 2 (Fun 𝐹 𝑥𝐴 (𝐹𝐵) = 𝑥𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹𝑦) ∈ 𝐵})
139, 12eqtr4d 2688 1 (Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wrex 2942  {crab 2945   ciun 4552  ccnv 5142  dom cdm 5143  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by: (None)
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