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Theorem unipreima 32400
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 6577 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 r19.42v 3185 . . . . . . 7 (∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥))
32bicomi 223 . . . . . 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 4907 . . . . . . 7 ((𝐹𝑦) ∈ 𝐴 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥)
65anbi2i 622 . . . . . 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 7061 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
98rexbidv 3173 . . . . 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 7061 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴)))
12 eliun 4995 . . . . 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 2725 . 2 (𝐹 Fn dom 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
161, 15sylbi 216 1 (Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wrex 3065   cuni 4903   ciun 4991  ccnv 5671  dom cdm 5672  cima 5675  Fun wfun 6536   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  imambfm  33805  dstrvprob  34014
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