Theorem unima 6903

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 1136	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐹 Fn 𝐴)
2		simpl 482	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	unssd 4141	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
5	4	3adant1 1130	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
6	1, 5	fvelimabd 6901	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ ∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦))
7		rexun 4145	. . . . 5 ⊢ (∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦 ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
8	6, 7	bitrdi 287	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦)))
9		fvelimab 6900	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
10	9	3adant3 1132	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
11		fvelimab 6900	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
12	11	3adant2 1131	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
13	10, 12	orbi12d 918	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → ((𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦)))
14	8, 13	bitr4d 282	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶))))
15		elun 4102	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)))
16	14, 15	bitr4di 289	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ 𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶))))
17	16	eqrdv 2731	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3057   ∪ cun 3896   ⊆ wss 3898   “ cima 5622   Fn wfn 6481  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  cycpmco2rn  33101  icccncfext  46009