Theorem fun 5063

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 5005	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
2	1	expcom 109	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
3		rnun 4732	. . . . . 6 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
4		unss12 3115	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷))
5	3, 4	syl5eqss 2989	. . . . 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 318	. . 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 4906	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
9		df-f 4906	. . . . 5 ⊢ (𝐺:𝐵⟶𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷))
10	8, 9	anbi12i 433	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)))
11		an4 520	. . . 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 173	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
13		df-f 4906	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
14	7, 12, 13	3imtr4g 194	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)))
15	14	impcom 116	1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∪ cun 2915   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  ran crn 4346   Fn wfn 4897  ⟶wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by:  fun2  5064  ftpg  5347  fsnunf  5362