Theorem unima 6960

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 1154	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐹 Fn 𝐴)
2		simpl 488	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3		simpr 490	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	unssd 4190	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
5	4	3adant1 1148	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
6	1, 5	fvelimabd 6958	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ ∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦))
7		rexun 4194	. . . . 5 ⊢ (∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦 ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
8	6, 7	bitrdi 290	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦)))
9		fvelimab 6957	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
10	9	3adant3 1150	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
11		fvelimab 6957	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
12	11	3adant2 1149	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
13	10, 12	orbi12d 932	. . . 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 285	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶))))
15		elun 4152	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)))
16	14, 15	bitr4di 292	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ 𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶))))
17	16	eqrdv 2758	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149   “ cima 5653   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  cycpmco2rn  33732  icccncfext  46990