Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unima Structured version   Visualization version   GIF version

Theorem unima 6720
 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 1133 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → 𝐹 Fn 𝐴)
2 simpl 486 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐵𝐴)
3 simpr 488 . . . . . . . 8 ((𝐵𝐴𝐶𝐴) → 𝐶𝐴)
42, 3unssd 4146 . . . . . . 7 ((𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
543adant1 1127 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ⊆ 𝐴)
61, 5fvelimabd 6719 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ ∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦))
7 rexun 4150 . . . . 5 (∃𝑥 ∈ (𝐵𝐶)(𝐹𝑥) = 𝑦 ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
86, 7syl6bb 290 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
9 fvelimab 6718 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
1093adant3 1129 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
11 fvelimab 6718 . . . . . 6 ((𝐹 Fn 𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
12113adant2 1128 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 (𝐹𝑥) = 𝑦))
1310, 12orbi12d 916 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → ((𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)) ↔ (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ∨ ∃𝑥𝐶 (𝐹𝑥) = 𝑦)))
148, 13bitr4d 285 . . 3 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶))))
15 elun 4109 . . 3 (𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶)) ↔ (𝑦 ∈ (𝐹𝐵) ∨ 𝑦 ∈ (𝐹𝐶)))
1614, 15syl6bbr 292 . 2 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝑦 ∈ (𝐹 “ (𝐵𝐶)) ↔ 𝑦 ∈ ((𝐹𝐵) ∪ (𝐹𝐶))))
1716eqrdv 2822 1 ((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ (𝐵𝐶)) = ((𝐹𝐵) ∪ (𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∃wrex 3133   ∪ cun 3916   ⊆ wss 3918   “ cima 5539   Fn wfn 6331  ‘cfv 6336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344 This theorem is referenced by:  cycpmco2rn  30785  icccncfext  42371
 Copyright terms: Public domain W3C validator