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Theorem unima 39168
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 1060 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → 𝐹 Fn 𝐴)
2 simpl 473 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐵𝐴)
3 simpr 477 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐶𝐴)
42, 3unssd 3787 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
543adant1 1078 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
6 fvelimab 6251 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝐵𝐶) ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
71, 5, 6syl2anc 693 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
8 rexun 3791 . . . . 5 (∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦 ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
97, 8syl6bb 276 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
10 fvelimab 6251 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
11103adant3 1080 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
12 fvelimab 6251 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
13123adant2 1079 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
1411, 13orbi12d 746 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
159, 14bitr4d 271 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶))))
16 elun 3751 . . 3 (𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)))
1715, 16syl6bbr 278 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ 𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶))))
1817eqrdv 2619 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1037   = wceq 1482  wcel 1989  wrex 2912  cun 3570  wss 3572  cima 5115   Fn wfn 5881  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894
This theorem is referenced by:  icccncfext  39869
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