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Theorem unima 38238
Description: Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
unima ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))

Proof of Theorem unima
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1053 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → 𝐹 Fn 𝐴)
2 simpl 471 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐵𝐴)
3 simpr 475 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐶𝐴)
42, 3unssd 3655 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
543adant1 1071 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
6 fvelimab 6047 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐵𝐶) ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
71, 5, 6syl2anc 690 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
8 rexun 3659 . . . . 5 (∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦 ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
97, 8syl6bb 274 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
10 fvelimab 6047 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
11103adant3 1073 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
12 fvelimab 6047 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
13123adant2 1072 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
1411, 13orbi12d 741 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
159, 14bitr4d 269 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶))))
16 elun 3619 . . 3 (𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)))
1715, 16syl6bbr 276 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ 𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶))))
1817eqrdv 2512 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1938  wrex 2801  cun 3442  wss 3444  cima 4935   Fn wfn 5684  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-fv 5697
This theorem is referenced by:  icccncfext  38673
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