Theorem fun 6066

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 5997	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
2	1	expcom 451	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
3		rnun 5541	. . . . . 6 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
4		unss12 3785	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷))
5	3, 4	syl5eqss 3649	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))
6	5	a1i 11	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
7	2, 6	anim12d 586	. . 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 5892	. . . . 5 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
9		df-f 5892	. . . . 5 ⊢ (𝐺:𝐵⟶𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷))
10	8, 9	anbi12i 733	. . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)))
11		an4 865	. . . 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 264	. . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
13		df-f 5892	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
14	7, 12, 13	3imtr4g 285	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)))
15	14	impcom 446	1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ran crn 5115   Fn wfn 5883  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fun2  6067  ftpg  6423  fsnunf  6451  ralxpmap  7907  hashfxnn0  13124  hashfOLD  13126  cats1un  13475  pwssplit1  19059  axlowdimlem10  25831  wlkp1  26578  padct  29497  eulerpartlemt  30433  sseqf  30454  poimirlem3  33412  poimirlem16  33425  poimirlem19  33428  poimirlem22  33431  poimirlem23  33432  poimirlem24  33433  poimirlem25  33434  poimirlem28  33437  poimirlem29  33438  poimirlem31  33440  mapfzcons  37279  diophrw  37322  diophren  37377  pwssplit4  37659  aacllem  42547