Theorem foun 6835

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 6790	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
2		fofn 6790	. . . 4 ⊢ (𝐺:𝐶–onto→𝐷 → 𝐺 Fn 𝐶)
3	1, 2	anim12i 625	. . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) → (𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶))
4		fnun 6645	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
5	3, 4	sylan 592	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
6		rnun 6134	. . 3 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
7		forn 6791	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
8	7	ad2antrr 739	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐹 = 𝐵)
9		forn 6791	. . . . 5 ⊢ (𝐺:𝐶–onto→𝐷 → ran 𝐺 = 𝐷)
10	9	ad2antlr 740	. . . 4 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran 𝐺 = 𝐷)
11	8, 10	uneq12d 4116	. . 3 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵 ∪ 𝐷))
12	6, 11	eqtrid 2808	. 2 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → ran (𝐹 ∪ 𝐺) = (𝐵 ∪ 𝐷))
13		df-fo 6537	. 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–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∪ cun 3897   ∩ cin 3898  ∅c0 4279  ran crn 5652   Fn wfn 6526  –onto→wfo 6529

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537

This theorem is used by: (None)