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Theorem unipreima 29574
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 5956 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 r19.42v 3121 . . . . . . 7 (∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥))
32bicomi 214 . . . . . 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 4472 . . . . . . 7 ((𝐹𝑦) ∈ 𝐴 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥)
65anbi2i 730 . . . . . 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 6377 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
98rexbidv 3081 . . . . 5 (𝐹 Fn dom 𝐹 → (∃𝑥𝐴 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
104, 7, 93bitr4d 300 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
11 elpreima 6377 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴)))
12 eliun 4556 . . . . 5 (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥))
1312a1i 11 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
1410, 11, 133bitr4d 300 . . 3 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ 𝑦 𝑥𝐴 (𝐹𝑥)))
1514eqrdv 2649 . 2 (𝐹 Fn dom 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
161, 15sylbi 207 1 (Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942   cuni 4468   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:  imambfm  30452  dstrvprob  30661
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