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Theorem unipreima 29776
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 6079 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 r19.42v 3230 . . . . . . 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 4592 . . . . . . 7 ((𝐹𝑦) ∈ 𝐴 ↔ ∃𝑥𝐴 (𝐹𝑦) ∈ 𝑥)
65anbi2i 732 . . . . . 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 6501 . . . . . 6 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹𝑥) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝑥)))
98rexbidv 3190 . . . . 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 6501 . . . 4 (𝐹 Fn dom 𝐹 → (𝑦 ∈ (𝐹 𝐴) ↔ (𝑦 ∈ dom 𝐹 ∧ (𝐹𝑦) ∈ 𝐴)))
12 eliun 4676 . . . . 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 2758 . 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 1632  wcel 2139  wrex 3051   cuni 4588   ciun 4672  ccnv 5265  dom cdm 5266  cima 5269  Fun wfun 6043   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  imambfm  30654  dstrvprob  30863
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