Theorem foun 6848

Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
foun	⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷))

Proof of Theorem foun

Step	Hyp	Ref	Expression
1		fofn 6804	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 6804	. . . 4 ⊢ (𝐺:𝐶–onto→𝐷 → 𝐺 Fn 𝐶)
3	1, 2	anim12i 613	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶))
4		fnun 6660	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
5	3, 4	sylan 580	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
6		rnun 6142	. . 3 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
7		forn 6805	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
8	7	ad2antrr 724	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐹 = 𝐵)
9		forn 6805	. . . . 5 ⊢ (𝐺:𝐶–onto→𝐷 → ran 𝐺 = 𝐷)
10	9	ad2antlr 725	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐺 = 𝐷)
11	8, 10	uneq12d 4163	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵 ∪ 𝐷))
12	6, 11	eqtrid 2784	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷))
13		df-fo 6546	. 2 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷)))
14	5, 12, 13	sylanbrc 583	1 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∪ cun 3945   ∩ cin 3946  ∅c0 4321  ran crn 5676   Fn wfn 6535  –onto→wfo 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546

This theorem is referenced by: (None)