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Theorem unipreima 32659
Description: Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
unipreima (Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem unipreima
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 funfn 6597 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 r19.42v 3188 . . . . . . 7 (∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥))
32bicomi 224 . . . . . 6 ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥) ↔ ∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥))
43a1i 11 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥) ↔ ∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
5 eluni2 4915 . . . . . . 7 ((𝐹𝑦) ∈ 𝐴 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥)
65anbi2i 623 . . . . . 6 ((𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥))
76a1i 11 . . . . 5 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥)))
8 elpreima 7077 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
98rexbidv 3176 . . . . 5 (𝐹 Fn dom 𝐹 → (∃𝑥𝐴 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
104, 7, 93bitr4d 311 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
11 elpreima 7077 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴)))
12 eliun 4999 . . . . 5 (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥))
1312a1i 11 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
1410, 11, 133bitr4d 311 . . 3 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ 𝑦 𝑥𝐴 (𝐹𝑥)))
1514eqrdv 2732 . 2 (𝐹 Fn dom 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
161, 15sylbi 217 1 (Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067   cuni 4911   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-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-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:  imambfm  34243  dstrvprob  34452
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