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Theorem unpreima 7062
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 6576 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elpreima 7057 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵))))
3 elun 4148 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴𝐵) ↔ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵))
43anbi2i 624 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)))
5 andi 1007 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
64, 5bitri 275 . . . . 5 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
7 elun 4148 . . . . . 6 (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ (𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)))
8 elpreima 7057 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
9 elpreima 7057 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
108, 9orbi12d 918 . . . . . 6 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
117, 10bitrid 283 . . . . 5 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
126, 11bitr4id 290 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
132, 12bitrd 279 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
1413eqrdv 2731 . 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 397  wo 846   = wceq 1542  wcel 2107  cun 3946  ccnv 5675  dom cdm 5676  cima 5679  Fun wfun 6535   Fn wfn 6536  cfv 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549
This theorem is referenced by:  sibfof  33328
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