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 1142	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐹 Fn 𝐴)
2		simpl 483	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3		simpr 485	. . . . . . . 8 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → 𝐶 ⊆ 𝐴)
4	2, 3	unssd 4128	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐵 ∪ 𝐶) ⊆ 𝐴)
5	4	3adant1 1136	. . . . . 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 4132	. . . . 5 ⊢ (∃𝑥 ∈ (𝐵 ∪ 𝐶)(𝐹‘𝑥) = 𝑦 ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ∨ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
8	6, 7	bitrdi 288	. . . 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 1138	. . . . 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 1137	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 (𝐹‘𝑥) = 𝑦))
13	10, 12	orbi12d 924	. . . 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 283	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶))))
15		elun 4090	. . 3 ⊢ (𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)) ↔ (𝑦 ∈ (𝐹 “ 𝐵) ∨ 𝑦 ∈ (𝐹 “ 𝐶)))
16	14, 15	bitr4di 290	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ (𝐵 ∪ 𝐶)) ↔ 𝑦 ∈ ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶))))
17	16	eqrdv 2738	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ (𝐵 ∪ 𝐶)) = ((𝐹 “ 𝐵) ∪ (𝐹 “ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3064   ∪ cun 3888   ⊆ wss 3890   “ cima 5628   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 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  cycpmco2rn  33213  icccncfext  46337