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Theorem unpreima 5292
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 4931 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elpreima 5286 . . . 4 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵))))
3 elun 3084 . . . . . 6 (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ (𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)))
4 elpreima 5286 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴)))
5 elpreima 5286 . . . . . . 7 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
64, 5orbi12d 707 . . . . . 6 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐹𝐴) ∨ 𝑥 ∈ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
73, 6syl5bb 181 . . . . 5 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵))))
8 elun 3084 . . . . . . 7 ((𝐹𝑥) ∈ (𝐴𝐵) ↔ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵))
98anbi2i 430 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)))
10 andi 731 . . . . . 6 ((𝑥 ∈ dom 𝐹 ∧ ((𝐹𝑥) ∈ 𝐴 ∨ (𝐹𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
119, 10bitri 173 . . . . 5 ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ 𝐵)))
127, 11syl6rbbr 188 . . . 4 (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹𝑥) ∈ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
132, 12bitrd 177 . . 3 (𝐹 Fn dom 𝐹 → (𝑥 ∈ (𝐹 “ (𝐴𝐵)) ↔ 𝑥 ∈ ((𝐹𝐴) ∪ (𝐹𝐵))))
1413eqrdv 2038 . 2 (𝐹 Fn dom 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
151, 14sylbi 114 1 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wo 629   = wceq 1243  wcel 1393  cun 2915  ccnv 4344  dom cdm 4345  cima 4348  Fun wfun 4896   Fn wfn 4897  cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
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