Theorem fun 5387

Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Assertion

Ref	Expression
fun	⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))

Proof of Theorem fun

Step	Hyp	Ref	Expression
1		fnun 5321	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
2	1	expcom 116	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
3		rnun 5036	. . . . . 6 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
4		unss12 3307	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷))
5	3, 4	eqsstrid 3201	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))
6	5	a1i 9	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
7	2, 6	anim12d 335	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))))
8		df-f 5219	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
9		df-f 5219	. . . . 5 ⊢ (𝐺:𝐵⟶𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷))
10	8, 9	anbi12i 460	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)))
11		an4 586	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
12	10, 11	bitri 184	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
13		df-f 5219	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
14	7, 12, 13	3imtr4g 205	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)))
15	14	impcom 125	1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∪ cun 3127   ∩ cin 3128   ⊆ wss 3129  ∅c0 3422  ran crn 4626   Fn wfn 5210  ⟶wf 5211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-fun 5217  df-fn 5218  df-f 5219

This theorem is referenced by:  fun2  5388  ftpg  5699  fsnunf  5715