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Theorem unpreima 5408
Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
unpreima (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))

Proof of Theorem unpreima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funfn 5031 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elpreima 5402 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵))))
3 elun 3139 . . . . . 6 (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ (𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)))
4 elpreima 5402 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
5 elpreima 5402 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
64, 5orbi12d 742 . . . . . 6 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
73, 6syl5bb 190 . . . . 5 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
8 elun 3139 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴𝐵) ↔ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵))
98anbi2i 445 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)))
10 andi 767 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
119, 10bitri 182 . . . . 5 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
127, 11syl6rbbr 197 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
132, 12bitrd 186 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
1413eqrdv 2086 . 2 (𝐹 Fn dom 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
151, 14sylbi 119 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 664   = wceq 1289  wcel 1438  cun 2995  ccnv 4427  dom cdm 4428  cima 4431  Fun wfun 4996   Fn wfn 4997  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by: (None)
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