Theorem unima 38340

Description: Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
unima	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))

Proof of Theorem unima

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1054	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐹 Fn 𝐴)
2		simpl 472	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3		simpr 476	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	unssd 3751	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
5	4	3adant1 1072	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
6		fvelimab 6163	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∪ 𝐶) ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ ∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦))
7	1, 5, 6	syl2anc 691	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ ∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦))
8		rexun 3755	. . . . 5 ⊢ (∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦 ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
9	7, 8	syl6bb 275	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦)))
10		fvelimab 6163	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
11	10	3adant3 1074	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
12		fvelimab 6163	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
13	12	3adant2 1073	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
14	11, 13	orbi12d 742	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦)))
15	9, 14	bitr4d 270	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶))))
16		elun 3715	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)))
17	15, 16	syl6bbr 277	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ 𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶))))
18	17	eqrdv 2608	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))

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