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Theorem unpreima 7063
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 6577 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elpreima 7058 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵))))
3 elun 4147 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴𝐵) ↔ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵))
43anbi2i 621 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)))
5 andi 1004 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
64, 5bitri 274 . . . . 5 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
7 elun 4147 . . . . . 6 (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ (𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)))
8 elpreima 7058 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
9 elpreima 7058 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
108, 9orbi12d 915 . . . . . 6 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
117, 10bitrid 282 . . . . 5 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
126, 11bitr4id 289 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
132, 12bitrd 278 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
1413eqrdv 2728 . 2 (𝐹 Fn dom 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
151, 14sylbi 216 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 843   = wceq 1539  wcel 2104  cun 3945  ccnv 5674  dom cdm 5675  cima 5678  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  sibfof  33637
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