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Theorem unima 38340
 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 1054 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → 𝐹 Fn 𝐴)
2 simpl 472 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐵𝐴)
3 simpr 476 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐶𝐴)
42, 3unssd 3751 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
543adant1 1072 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
6 fvelimab 6163 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐵𝐶) ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
71, 5, 6syl2anc 691 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
8 rexun 3755 . . . . 5 (∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦 ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
97, 8syl6bb 275 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
10 fvelimab 6163 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
11103adant3 1074 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
12 fvelimab 6163 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
13123adant2 1073 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
1411, 13orbi12d 742 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
159, 14bitr4d 270 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶))))
16 elun 3715 . . 3 (𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)))
1715, 16syl6bbr 277 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ 𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶))))
1817eqrdv 2608 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540   “ cima 5041   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  icccncfext  38773
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