Theorem foun 6868

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⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)⟶onto(𝐵 ∪ 𝐷))

Proof of Theorem foun

Step	Hyp	Ref	Expression
1		fofn 6823	. . . 4 ⊢ (𝐹:𝐴⟶onto𝐵 → 𝐹 Fn 𝐴)
2		fofn 6823	. . . 4 ⊢ (𝐺:𝐶⟶onto𝐷 → 𝐺 Fn 𝐶)
3	1, 2	anim12i 625	. . 3 ⊢ ((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶))
4		fnun 6678	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
5	3, 4	sylan 592	. 2 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
6		rnun 6149	. . 3 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
7		forn 6824	. . . . 5 ⊢ (𝐹:𝐴⟶onto𝐵 → ran 𝐹 = 𝐵)
8	7	ad2antrr 739	. . . 4 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐹 = 𝐵)
9		forn 6824	. . . . 5 ⊢ (𝐺:𝐶⟶onto𝐷 → ran 𝐺 = 𝐷)
10	9	ad2antlr 740	. . . 4 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐺 = 𝐷)
11	8, 10	uneq12d 4171	. . 3 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵 ∪ 𝐷))
12	6, 11	eqtrid 2810	. 2 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷))
13		df-fo 6567	. 2 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)⟶onto(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷)))
14	5, 12, 13	sylanbrc 595	1 ⊢ (((𝐹:𝐴⟶onto𝐵 ∧ 𝐺:𝐶⟶onto𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)⟶onto(𝐵 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ran crn 5657   Fn wfn 6556  ⟶ontowfo 6559

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567

This theorem is used by: (None)