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Theorem unipreima 30077
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 6262 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 r19.42v 3313 . . . . . . 7 (∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥))
32bicomi 225 . . . . . 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 4755 . . . . . . 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 6700 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
98rexbidv 3262 . . . . 5 (𝐹 Fn dom 𝐹 → (∃𝑥𝐴 𝑦 ∈ (𝐹𝑥) ↔ ∃𝑥𝐴 (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
104, 7, 93bitr4d 312 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
11 elpreima 6700 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴)))
12 eliun 4835 . . . . 5 (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥))
1312a1i 11 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 𝑥𝐴 (𝐹𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐹𝑥)))
1410, 11, 133bitr4d 312 . . 3 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ 𝑦 𝑥𝐴 (𝐹𝑥)))
1514eqrdv 2795 . 2 (𝐹 Fn dom 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
161, 15sylbi 218 1 (Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wrex 3108   cuni 4751   ciun 4831  ccnv 5449  dom cdm 5450  cima 5453  Fun wfun 6226   Fn wfn 6227  cfv 6232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-fv 6240
This theorem is referenced by:  imambfm  31133  dstrvprob  31342
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