Theorem fun 5295

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 5229	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
2	1	expcom 115	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
3		rnun 4947	. . . . . 6 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
4		unss12 3248	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷))
5	3, 4	eqsstrid 3143	. . . . 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 333	. . 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 5127	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
9		df-f 5127	. . . . 5 ⊢ (𝐺:𝐵⟶𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷))
10	8, 9	anbi12i 455	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)))
11		an4 575	. . . 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 183	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
13		df-f 5127	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
14	7, 12, 13	3imtr4g 204	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)))
15	14	impcom 124	1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  ran crn 4540   Fn wfn 5118  ⟶wf 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127

This theorem is referenced by:  fun2  5296  ftpg  5604  fsnunf  5620