Theorem unima 6909

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 4144	. . . . . . 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 6907	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ ∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦))
7		rexun 4148	. . . . 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 6906	. . . . . 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 6906	. . . . . 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 4105	. . 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 2734	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 3060   ∪ cun 3899   ⊆ wss 3901   “ cima 5627   Fn wfn 6487  ‘cfv 6492

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  cycpmco2rn  33207  icccncfext  46127